6 Mathematical models Suppose ou are given a problem it ma be a practical question that a friend asks ou, or an issue that ou have noticed ourself to which ou want to find a solution; it ma even be a mathematical problem that ou are set in one of our classes. How do ou solve it? If it is a practical problem, ou might tr out different approaches or talk it through with a friend. If it is a problem encountered in class, ou might go back over our notes and tetbooks, looking for a similar eample to see how that was solved. The mathematician George Póla wrote a book called How to Solve It, which was published in 1945. In this book he describes several distinct steps taken b people who are good at problem-solving. These steps are: understand the problem make a plan carr out our plan look back at our solution to see whether ou can improve it or use it in another contet. This process of solving problems is called mathematical modelling, and the stages outlined b Póla constitute what is known as the modelling ccle. Who u ses mathematical modelling? It is the normal wa for mathematicians to work, as well as designers, engineers, architects, doctors and man other professionals. If ou become good at problem-solving, ou can reduce the number of things that ou have to memorise: it is easier to remember formulae or concepts when ou understand how the work. Prior learning topics It will be easier to stud this topic if ou have completed: Chapter Chapter 14. 49 Topic 6 Mathematical models
Chapter 17 Functions and graphs Mathematical models allow us to use mathematics to solve practical problems, describe different aspects of the real world, test ideas, and make predictions about the future. A model ma be a simplification of a real situation, but it is usuall quick and cheap to work with and helps us to strengthen our understanding of the problem. The concept of a function is fundamental to mathematical modelling. Before ou can work with the more comple models in this topic, ou need to be confident in our understanding and use of functions and function notation. 17.1 What is a function? A function is like a mapping diagram between two sets, where each element of one set maps to a single element of the other set. A function can also be viewed as a machine where ou feed numbers in at one end, it works on the numbers according to a certain rule, and then gives ou the output at the other end. For eample: In this chapter ou will learn: what a function is and the definitions of domain and range about the graphs of some basic functions how to use function notation how to read and interpret a graph and use it to find the domain and range of a function about simple rational functions and asmptotes how to draw accurate graphs how to create a sketch from given information how to use our GDC to solve equations that involve combinations of the basic functions dealt with in this course. 5 4 1 5 9 5 1 1 + 9 6 6 4 Wh can we use maths to describe real-world situations and make predictions? Are we imposing our own mathematical models on the world, or is it that we are discovering that the world runs according to the rules of mathematics? 1 Domain Range The set of numbers that go into the function is called the domain. Values in the domain are: the numbers chosen to describe that particular function values taken b the independent variable the numbers plotted on the horizontal ais when ou draw a graph of that function. You learned about dependent and independent variables in Chapter 1. 17 Functions and graphs 493
hint Be careful not to confuse this use of the word range with its use in the contet of statistical data in Chapter 7. The numbers that come out of the function make up the range of that function. The are: values taken b the dependent variable the numbers plotted on the vertical ais when ou draw a graph of the function. Here are some alternative was of thinking about a function: a mapping diagram a set of ordered pairs a graph an algebraic epression. When representing a function b an of the above methods, ou have to be careful because not all mapping diagrams, sets of ordered pairs, graphs or algebraic epressions describe a valid function. The important point to remember is that a valid function can have onl one output for each input value that goes into the function. Mapping diagram These are functions These are not functions Eplanation 4 6 3 7 11 4 6 5 7 In the first diagram there is one output for ever input. In the second diagram there are two different outputs for the input 6. Domain Range Domain Range Set of ordered pairs {(6, 8), (3, ), (5, 9), ( 1, )} The domain consists of the -coordinates: {6, 3, 5, 1} The range consists of the -coordinates: {8,, 9, } 6 4 {(6, 8) (3, ) (6, 9) ( 1, )} The domain consists of the -coordinates: {6, 3, 6, 1} The range consists of the -coordinates: {8,, 9, } 3 1 The first set of number pairs has one -coordinate for ever -coordinate. The second set has two different -coordinates for the -coordinate 6. The first graph has one -value for ever -value. The second graph has two -values for each positive -value. Graph 3 1 1 3 1 1 1 3 4 6 3 table continues on net page... 494 Topic 6 Mathematical models
continued... These are functions These are not functions Eplanation 6 4 1 1 4 This is a function 3 4 This is not a function 5 Graphs continued... This suggests the vertical line test : if a vertical line drawn at an position in the domain crosses the graph onl once, then the graph represents a valid function. 6 Algebraic epression f() = 3 + f ) = ± The first epression will give 4 the blue curve in the graph above. Plotting the second epression will give a circle, which fails the vertical line test. The f() notation is eplained further in section 17.. Eercise 17.1 1. For each of the graphs below, indicate whether or not the graph represents a valid function. (a) 6 4 (b) 6 4 6 4 4 6 4 6 4 4 6 4 (c) 1 6 (d) 6 6 5 4 1 5 5 1 5 1 4 4 6 8 1 4 6 17 Functions and graphs 495
(e) 1 (f) 1 8 8 6 6 4 4 4 4 6 8 1 4 4 6 4 17. Functions in more detail Function notation If ou are asked to draw a graph of = 3 + 5, ou know that this will be a curve because the powers of are greater than 1, suggesting it is not linear. You can use the equation to calculate the values of that correspond to particular values of. For instance, if = 3, then: = 3 3 + 3 5 = 7 + 9 5 = 31 So the point (3, 31) lies on the curve. The mathematical relationship = 3 + 5 can also be written in function notation f() = 3 + 5. The second form gives ou the same information so wh do we need this different notation for epressing the same thing? The equation = 3 + 5 tells ou the link between the - and -coordinates of a graph, which enables ou to draw that graph. The function notation f() = 3 + 5 epresses the idea of a function as a machine : f f() So represents the input, f is the machine, and f() is the output (the result of appling f to ). The function notation f(), pronounced as f of, allows us to: distinguish between different functions b using different letters; for eample, f(), g(), h() 496 Topic 6 Mathematical models
use an efficient shorthand for substituting values; for eample, f(3) means replace b 3 so that f(3) = 3 3 + 3 5 = 7 + 9 5 = 31 match a formula to a phsical quantit in a meaningful wa; for eample, we can write h(t) = 5t + 3t 1 to show that this is a formula to find the height at a given time. Worked eample 17.1 Q. (a) If f() = 5 3, find f(4), f( ) and f(). (b) If V() r = 4 3 π r, find V(), V(1) and V(1) in terms of π. 3 (c) If f() = 3 +, find when f() = 4. f(4), f( ) and f() mean to replace in the epression 5 3 with 4, and, respectivel. V(), V(1) and V(1) mean to replace in the epression 4 3 3 πr b, 1 and 1, respectivel. (d) If f() = 3 + + 5, find a simplified epression for f(). A. (a) f(4) = 5 3 4 = 7 f( ) = 5 3 ( ) = 11 f() = 5 3 = 5 (b) V() = 4 3 π 3 = 3 π 3 V(1) = 4 3 π 13 = 34π f() = 4 is the same as 3 + = 4, which is a quadratic equation that ou can solve using our chosen method from Chapter. V(1) = 4 3 π 13 = 1348π (c) 3 + = 4 3 4 = TEXAS CASIO f() means to replace ever occurrence of in the epression 3 + + 5 b. It is best to put brackets around, so that ou will remember to cube and square the as well as. From GDC: = 5 or = 8 (d) f() = () 3 () + () + 5 = () () () () () + () + 5 = 8 3 8 + + 5 17 Functions and graphs 497
Eercise 17. 1. A function is defined as f() = 3 8. Find the following: 1 (a) f(4) (b) f( 1) (c) f (d) f(a) 3. Given the function g() = 11 5, find: (a) g(6) (b) g(1) (c) g(7) + 13 (d) g(a) 3. Given the function f() = 7 + 5, find: (a) f( ) (b) f() (c) f(3) 5 (d) f(c) 4. Given that g() = 3 4 + 3 7, find: (a) g( 4) (b) g( 1) (c) 13g(5) 5. For the function h(t) = 14t 4.9t, find: (a) h(1) (b) h 1 7 6. A function is defined as f() = + for. (a) Work out f( 7). 3 (b) Find f. 4 (c) Calculate f(6) f(). (d) Find a simplified epression for f( + 1). Dom ain and range (c) h(3) h() (d) 5h(1) + If ou are drawing a graph, the domain of a function consists of the values that are plotted along the -ais; these are values of the independent variable. The domain ma be all the real numbers on a number line, or it ma be a restricted interval such as 3. Suppose that a function is defined b f() = 3. If the domain is not specified, there are no values of that cannot be used, and the graph looks like this: 8 6 4 6 4 4 6 8 4 Note the arrows at both ends of the line. 498 Topic 6 Mathematical models
If the domain is given as R, 1, the graph looks like this: 8 6 4 3 1 1 3 4 4 You saw in Chapter 1 that R is the smbol for real numbers and in Chapter 8 that is set notation for is a member of ; see Chapter 1 if ou need to revise set notation for the various tpes of numbers and Chapter 8 if ou need a reminder of set notation. Note that there is an arrow at onl one end of the line; the end at = 1 is marked b a filled circle. R, 1 indicates that the domain is all real numbers greater than or equal to 1. If the domain is given as 3, the graph looks like this: 8 6 4 3 1 1 3 4 4 Now both ends, at = and = 3, are marked b filled circles. 17 Functions and graphs 499
If the value of one of the ends is not included in the domain, that end would be marked with an empt (non-filled) circle. For eample, if the domain were < 3 (which means = is not included in the domain), then the graph would look like this: 8 6 4 3 1 1 3 4 4 The range also changes depending on the domain: For f() = 3, R, the range is also f() R. For f() = 3, R, 1, the range is f() 1, as ou can see from the first graph on page 499. For f() = 3, R, 3, the range is 3 f() 7, as ou can see from the second graph on page 499. Worked eample 17. Q. Write down the do main and range for each of these functions: (a) 4 3 1 1 3 4 6 8 5 Topic 6 Mathematical models
continued... (b) 6 4 3 1 1 3 4 6 (c) 3 1 1 1 3 The graph has fi lled circles at both ends, so the domain is restricted between, and also includes, the -coordinates of these ends. The range is between the -coordinates of these ends. 1 A. (a) The domain is 3 3. The range is 7 f().5. The graph has arrows at both ends, so the domain is all real numbers. The curve has a maimum point at (, ), so the range is all numbers less than or equal to. (b) The domain is R. The range is f(). There is an arrow at one end of the curve, which means that the domain (and range) is unrestricted in this direction; a fi lled circle at the other end means that the domain reaches up to = but no further. The minimum -value on the curve is. (c) The domain is. The range is f(). If ou use our GDC to draw the graph of a function, ou can find the coordinates of important points on the curve, such as minimum and maimum points or the points at the beginning and end of the domain. See section.g(e) on page 648 or sections.3.17 and.3.18 on pages 678 683 of the GDC chapter for a reminder of the different methods. 17 Functions and graphs 51
Worked eample 17.3 Q. Use our GDC to draw graphs of the following functions. State the domain and range in each case. Plot the graph on our GDC and fi nd the coordinates of the minimum point on the parabola; this gives ou the lowest value of the range. There is no restriction given on the domain, so it consists of all real numbers. (a) f() = ( ) + 1 (b) f() = 3 ; (c) f() = 1 A. (a) TEXAS CASIO In this case the domain is restricted to values between, and including, and. Use our GDC to fi nd the coordinates of the points at the beginning and end of the domain. The minimum point on the parabola is at (, 1) The domain is R. The range is f() 1. (b) TEXAS CASIO The domain can take all real values, but the graph tells ou that values in the range are alwas above zero. (c) The domain is. The range is 6 f() 6. TEXAS CASIO You will learn about horizontal and vertical asmptotes in section 17.3. The domain is R. The range is f() >. The curve gets close to, but never touches, the -ais, so there is a horizontal asmptote at =. 5 Topic 6 Mathematical models
Eercise 17.3 1. Write down the domain and the range for each of the following functions. (a) 15 (b) 4 1 3 5 4 5 4 6 4 1 4 6 1 1 (c) 4 (d) 15 1 6 4 4 6 4 5 4 6 4 6 8 (e) (f) 5 1 4 4 4 6 3 1 1 3 1 5 5 1 4 1 17 Functions and graphs 53
. Use our GDC to draw the graphs of the following functions. State the range of each function. (a) = 4 7; 1 1 (b) = + 3; 7 < 7 (c) = 3 4; 5 4 (d) = ( + 5)( 1); 6 < < 6 17.3 Rationa l functions hint Compare this with the term rational number introduced in Chapter 1. A rational function is a function whose algebraic epression looks like a fraction or ratio. a The simplest form of a rational function is f ) = where a, b and b + c c are constants. The graph of such a function has a distinctive shape, known as a hperbola. In fact, the graph is made up of two curves; the values of a, b and c determine the positions of these curves. B using a GDC or a maths software package on the computer, ou can investigate how the constants a, b and c affect the basic shape of the graph. Be careful entering the function s formula: make sure to use brackets and division signs correctl. 3 For eample, to enter f() =, ou need brackets to tell the GDC 3 1 that it is dividing 3 b all of (3 + 1). If ou did not have the brackets around 3 + 1, the GDC ma draw the graph of f() = 3 + 1, which is a different function, as ou can see in the graph below. 3 6 6 f() = 3 + 1 3 4 4 4 6 f() = 3 3 + 1 4 vertical asmptote =.3333 6 Tr eploring hperbola graphs ourself. Here are some suggestions: a Start with the simplest rational function f ) = and draw graphs for different values of a. 54 Topic 6 Mathematical models
1 Draw graphs of f() = for different positive numbers c. ± c 1 Draw graphs of f() = + d for various values of d. ± c You will find that each graph has both a vertical and a horizontal break. For eample, the graph of f ) = + 1 looks like: 3 6 4 vertical asmptote = 3 6 4 f() = + 1 3 4 6 8 1 horizontal asmptote = 1 4 6 With our GDC, ou can use a trace function or the table of coordinates to find the approimate location of the breaks in the graph, which are called asmptotes. See sections 17. Finding a vertical asmptote and 17.3 Finding a horizontal asmptote on pages 679 and 68 of the GDC chapter if ou need to. TEXAS CASIO The horizontal break in the graph is called a horizontal asmptote. It occurs whenever a function s output () value approaches a certain number as increases or decreases. In the case of f ) = + 1, we can see that as increases, f() 3 approaches 1. The fraction gets smaller and smaller because 3 the denominator is getting larger and larger, so f() approaches 1 but will never actuall reach 1. Therefore, the equation of the horizontal asmptote is = 1. You can draw = 1 on our GDC to confirm that this is the location of the break. 17 Functions and graphs 55
TEXAS CASIO The vertical break in the graph is called a vertical asmptote. It occurs when a fraction s denominator becomes zero. a In general, the graph of f() = will have a vertical asmptote when ± c =. So: ± c the graph of f() = will have a vertical asmptote when + 3 + 3 =, i.e. at = 3 1 the graph of g() = will have a vertical asmptote when 5 5 =, i.e. at =.5. In both of the above cases, the horizontal asmptote is the line = (in other words, the -ais). To draw the graph of h ( ) = + 3, note that there will be a vertical 1 asmptote at = 1 and a horizontal asmptote at =. 6 horizontal asmptote = 4 6 4 f() = + 3 1 4 6 vertical asmptote = 1 4 6 Asmptotes can sometimes be difficult to identif on the GDC screen. If ou re having trouble locating the asmptotes on our GDC graph, check the table function for the list of coordinates of the points on the graph. (See 14.1 Accessing the table of coordinates from a plotted graph on page 678 of the GDC chapter if ou need to.) 56 Topic 6 Mathematical models
When identifing an asmptote from a table of coordinates: a horizontal asmptote is found b looking at what happens to the values as the values get larger: TEXAS CASIO a vertical asmptote occurs when the table shows an ERROR message: TEXAS CASIO Worked eample 17.4 Q. A function is defined b g() = 3 (a) Sketch its graph. 5 + for 6 3. Use our GDC to sketch the graph. (See.G Graphs on page 645 of the GDC chapter if ou need to.) (b) Write down the equation of the vertical asmptote. (c) What is the equation of the horizontal asmptote? A. (a) TEXAS CASIO 17 Functions and graphs 57
The denominator of the fraction equals zero when + =. From the GDC graph, it looks as though the horizontal asmptote is at = 3. Draw the line = 3 on our GDC to check this, or use the table of coordinates to look at the values as increases. See.G and/or 17.3 of the GDC chapters as appropriate. continued... (b) The vertical asmptote is =. (c) TEXAS The horizontal asmptote is = 3. CASIO Eercise 17.4 1. B drawing the graphs of the following functions, in each case: (i) state the equation of the vertical asmptote (ii) find the horizontal asmptote. (a) = 1 (e) = 1 1 (b) = 3 (f) = 4 3 (c) = 1 + 1 (g) = 5 (d) = + 1 7 3 (h) = 1 3 + 4 17.4 Drawing graphs and diagrams As ou have seen, graphs and diagrams are ver powerful tools. The can help ou to visualise and discover properties of the situation ou are studing. Before moving on to mathematical models in the net few chapters, ou need to: understand the difference between a sketch and an accurate graph be able to create a sketch from information that ou have been given be able to transfer a graph from our GDC onto paper know how to draw an accurate graph be able to draw combinations of two or more graphs on one diagram know how to use our graphs to solve equations that involve combinations of functions that ou have studied. 58 Topic 6 Mathematical models
T he difference between a sketch and an accurate graph It is important to understand the difference between the instructions plot, sketch and draw, and be able to choose the appropriate tpe of graph for a particular problem. Read an problem carefull and decide which kind of diagram will be the most useful initiall; ou can alwas draw another if the first does not give ou sufficient clarit. It is also important to make sure that an diagram ou draw is large enough! A tin diagram, squashed into one corner of the paper, will not help ou; nor will it be informative to the person who is checking or marking our work! Sketch Plot Draw A sketch represents a situation b means of a diagram or graph. It should be clearl labelled and give a general idea of the shape or relationship being described. All obvious and relevant features should be included, such as turning points, intercepts and asmptotes. The dimensions or points do not have to be drawn to an accurate scale/position. To plot a graph, ou need to calculate a set of accurate points and then mark them clearl, and accuratel, on our diagram using a correct scale. A drawing is a clearl labelled, accurate diagram or graph. It should be drawn to scale. Straight lines should be drawn with a ruler, and points that are known not to lie in a straight line should be joined b a smooth curve. When ou are drawing, plotting or sketching: Use a pencil ou ma need to make changes, and using permanent ink would make this difficult. Mark the aes for the graph and remember to label them. Not all graphs are graphs; for eample, ou might be plotting height against time. Alwas use the correct labels. Follow an instructions relating to the scale, the domain or the range of a graph. Does a graph without labels or scales still have meaning? Cr eating a sketch Draw the function f() = 4 1 on our GDC. What do ou notice? What are the most important features to include in a sketch? You should have obtained an image like this: TEXAS CASIO 17 Functions and graphs 59
The points to notice are: This is an graph, so the aes should be labelled and. The curve on the left of the -ais has a minimum at approimatel ( 1, 1). There is a vertical asmptote at = (the -ais). There is an -intercept between the points (1, ) and (, ). There is no -intercept. Did ou observe these features on our GDC graph? Now ou can make a sketch of the graph, b transferring these important features onto paper. 6 4 3 1 1 3 4 6 Draw the graph on our GDC. The curve is decreasing, so the value of the function at = is the maimum of the range, and the value at = 6 is the minimum of the range; these values can be found with our GDC. See section.g(e) The trace function on page 648 of the GDC chapter if ou need to. Worke d eample 17.5 Q. (a) Sketch the graph of f() = for the domain 6. Comment on the range of f(). A. (a) (b) Sketch the graph of h() = for the same domain as in (a). What do ou notice about the range of this function? TEXAS CASIO 51 Topic 6 Mathematical models
continued... Transfer the sketch from our GDC onto paper; make sure ou indicate the domain using fi lled circles at the ends of the curve. 1 1 1 3 4 5 6 1 The range is.449 f(). Draw the graph on our GDC. This graph has no values outside, and it is decreasing over this interval, so the maimum and minimum values are the values at = and =. (b) TEXAS CASIO Transfer the sketch from our GDC onto paper. Even though the graph has no values outside, ou should still make the -ais etend to = 6, so that it includes all of the given domain. 1 1 1 3 4 5 6 1 The range is f() 1.41. When gets bigger than, the quantit inside the square root becomes negative. It is impossible to find the square root of a negative number, so this has restricted the domain and range of (). 17 Functions and graphs 511
Plotting a graph Worked eample 17.6 To plot a graph, ou need to calculate the points that ou are going to put on the graph. You can do this b substituting values into the equation b hand or b using the table facilit on our GDC. (See 14.1 Accessing the table of coordinates from a plotted graph if ou need to.) Q. If f() = 3 + and g() = 3 1, plot f() and g() on the same graph. Use our graph to find the coordinates of the point where the two functions intersect. Take the domain to be. A. 1 1 f() = 3 + 1 1 3 16 1 1 g() = 3 1 7 4 1 5 Plot the points for each function. For f(), draw a curve through the plotted points; for g(), draw a line through the plotted points. 15 f() eam tip 1 When the solution is taken from a hand-drawn graph, as in Worked eample 17.6, it is diffi cult to be accurate to more than one decimal place. As long as our graph is clearl drawn with the intersection point marked, ou will be given credit for an answer within certain tolerances, e.g. ( 1.7 ±., 6.1 ±.). 1 5 5 1 1 g() You can see from the graph that there is onl one intersection. Estimate the coordinates of this intersection point. 15 The two functions cross at the point ( 1.7, 6.1). Eercise 17.5 1. Consider the function f() = 1. (a) Sketch f() for the domain 4 4. (b) Comment on the range of f(). (c) Let g() = 5 + f() for 4 4. Deduce the range of g(). Justif our answer. 51 Topic 6 Mathematical models
. (a) Sketch the graph of the function f() = 3 for the domain 1 3, and write down the range of f(). (b) Sketch the graph of the function g() = 3 1 3, and state the range of g(). (c) Sketch the graph of the function h() = 3 domain, and state the range of h(). 1 + 3 1 3 for the domain for the same (d) Using our results from above, deduce the range of g() + h(). Justif our answer. 3. For each of the following functions, use our GDC to draw the graph, then look at the main features of the graph and hence sketch it on paper. Take the domain of each function to be 3 3. (a) f() = 5 1 3 (b) f() = + 7 4 + 5 (c) f() = + 1 (d) f() = + 8 4. Plot the graphs of the following pairs of functions for the domain 3 3. In each case give the coordinates of the points where the two functions intersect to s.f. (a) f() = + 1, g() = 1 (b) f() =, g() = 1 (c) f() = 3 + 1, g() = 1 (d) f() = 5 1, g() = + 1 Solving equations using gr aphs on our GDC As ou saw in Chapter, there are standard algebraic methods for solving linear, simultaneous and quadratic equations. However, man equations are ver difficult to solve with algebra, and can often be solved much more easil b looking at a graph. Using a graph is also helpful because, as long as the domain and range are well chosen, it can show ou all the solutions to an equation. If ou use an equation solver on our GDC, ou might find onl one solution when there are several. To solve equations with a graph, draw the function on the left-hand side of the equals sign as one line or curve, and the function on the righthand side of the equation as another line or curve on the same graph; then look at where the intersect. The -coordinate of each intersection will be a root of the equation. 17 Functions and graphs 513
For eample, to solve the equation 3 3 + + 1 = 1 1, take the two functions f() = 3 3 + + 1 and g() = 1 1 and draw their graphs on the same aes. The graph of g() is a straight line, while the graph of f() is a cubic curve with two turning points. You might epect that the equation will have three solutions, but b drawing the actual graphs on the same set of aes ou will see that there is onl one intersection. You can use the intersection tool on our GDC to locate the point of intersection (see 19. (a) Solving unfamiliar equations using a graph on page 684 of the GDC chapter for a reminder of how to use this tool if ou need to). TEXAS CASIO GDC Remember that ou should round the answers given b the GDC to three signifi cant fi gures if no other specifi c degree of accurac has been requested. The GDC tells ou that the curves intersect at (.567, 1.8), so the solution to the equation 3 3 + + 1 = 1 1 is =.567. You can learn and practise this technique b plotting graphs on paper too. Worke d eample 17.7 Q. The functions f() and h() are defined as f() = + 1 and h() = 9 over the domain 4 4. (a) Fill in the missing values in the following table. 4 3 1 3 4 f() 1.65 1.5 9 h() 7 9 8 7 (b) Using a scale of cm to represent one unit on the -ais and 1 cm to represent two units on the -ais, draw both functions on the same graph. (c) Use our graph to find the solutions to the equation + 1 = 9. Give the answer to 1 d.p. 514 Topic 6 Mathematical models
continued... Substitute the given values of into the algebraic epressions for f() and h(). Label each curve. A. (a) 4 3 1 3 4 f() 1.65 1.15 1.5 3 9 17 h() 7 5 9 8 7 (b) Plot the points from (a) on a set of aes drawn according to the given scale. Then draw a curve through each set of points. Label each curve. 16 1 8 f() 4 4 4 4 Read off the -coordinates of the two points of intersection. It can be helpful to draw dashed lines from the points of intersection to the -ais. 8 h() (c) The solutions are =. and =.8. Plot the graphs of = + 1 and = 1 on our GDC. Be careful when ou enter the equations into our calculator. Use brackets for ( + 1) and ( ) to make sure that our GDC does the intended calculations. Worked eample 17.8 Q. Find the solutions of the equation A. TEXAS 1 + = 1. CASIO Find the -coordinates of the two points of intersection. (See 19. (a) Solving unfamiliar equations using a graph on page 684 of the GDC chapter for a reminder of how to use this tool if ou need to). The solutions are = 1.8 and = 1.8. 17 Functions and graphs 515
Eercise 17.6 1. The functions f() and g() are defined as f() = and g() = for 4. (a) Cop the following table and fill in the missing values. 1 1 3 4 f() 8 g() (b) Draw both functions on the same set of aes. (c) Use our graph to find the solutions to the equation =.. The functions f() and g() are defined as f() = 3 1 and g() = 5 for. (a) Cop the following table and fill in the missing values. 1 1 f() g() (b) Draw both functions on the same set of aes on graph paper. (c) Use our graph from (b) to find the solutions to the equation 3 1 = 5. Give our answer to 1 d.p. 3. The functions f() and g() are defined as f() = 3 and g() = 1 for 4. (a) Cop the following table and fill in the missing values. 4 3 1 1 f() 1.89 7 g() 1 7 (b) Draw both functions on the same set of aes on graph paper. (c) Use our graph to find the solutions to the equation 3 = 1. If our answer is not eact, give it to 1 d.p. 4. Let f() = 3 3 + 5 + and g() = 7 5. (a) Use our GDC to draw both functions on the same set of aes. (b) State the number of intersections of the two functions. (c) Hence find all the solutions of the equation f() = g() to 1 d.p. 516 Topic 6 Mathematical models
5. For each of the following pairs of functions, draw both functions on the same set of aes on our GDC, and hence: (i) state the number of intersections of the two functions (ii) find all the solutions to the equation f() = g(). (a) f() = 5 7 and g() = 3 +.451 (b) f() = +1 and g() = 3 (c) f() = 6.15 9 and g() =.78 1 (d) f()= 4 and g() = 3 (e) f() = 3 7 +1 and g() = 8 5 (f) f() = 3 and g() = 1 7 4 6. Solve the following equations with our GDC, using the graphical method. (a) = 3 + 6 (b) 3 3 + = 8 3 (c) 1.5 3 = 4 + 3.14 (d) 4 9 = + 6 (e) 4 + 5 +. = 4 3 + 3 1 (f) 1.7 +.875 1.4 = 1.5 4 + 4 3.5 (g) ( )( + 1)( + 4) =.73 + 1.96 7. Solve the following equations with our GDC, using the graphical method. (a) 1. t = 5 (b) 4.15 = 5.336 (c) 17.47t = 3 (d) 8 (5 +.336) = 43.17 (e) + 5 = 7 +.15 (f) 3.958 = 3 7 4 3 (g). 4 = 9. 645 1 (h) 1 3 = 63 5. 17 Functions and graphs 517
Summar You should kn ow: the concept of a function and the definitions of domain and range how to recognise and use function notation how to interpret the graph of a function and use it to find the domain and range how to recognise and identif simple rational functions and asmptotes how to draw or plot accurate graphs how to sketch a graph from information given how to use a GDC and the hand-drawn graphical method to solve equations that involve combinations of the functions studied in this course. 518 Topic 6 Mathematical models
Mied eamination pra ctice Eam-stle questions 1. A function is defined as f() = 7 13. (a) Find f(). (b) State the value of f(3). (c) Find an epression for: (i) f(a) (ii) f(a ). The graph of the function f() is shown below. 1 6 4 4 6 1 3 4 (a) Write down the domain of the function. (b) State the range of f(). (c) Given that f(k) =, find all the possible values of k. 3. Sketch the graphs of the following functions for 5 5. In each case: (i) State the equation of the vertical asmptote. (ii) Find the horizontal asmptote. (a) f() = 3 5 (b) g() = 4 + 9 7 17 Functions and graphs 519
4. Use our GDC to draw the graph of the function f() = 7 5 + 4, with domain 3 3. 1 8 Look at the main features of the graph and hence sketch the function on paper. 5. The functions f() and g() are defined as f() = 1 4 3 and g() = 7 for 3 3. (a) Cop the following table and fill in the missing values. 3 1 1 3 f() g() (b) Draw both functions on the same set of aes. (c) Use our graph from (b) to find the solutions to the equation 1 4 3 = 7. Give our answer to 1 d.p. 6. Two functions are defined as f() = 3 + 5 1 and g() = 4 + 3 + 6 4. Both functions have the same domain, 3 3. (a) Use our GDC to draw both functions on the same set of aes. (b) State the number of intersections of the two functions. (c) Hence find all the solutions of the equation f() = g() to 1 d.p. 5 Topic 6 Mathematical models
Chapter 18 Linear and quadratic models In the introduction to Topic 6, ou saw an outline of the steps of mathematical modelling and problem-solving described b the mathematician George Póla: 1. understand the problem. make a plan 3. carr out our plan 4. look back at our solution to see whether ou can improve it or use it in another contet. Understanding the problem can take time. You need to clarif what ou need to find or show, draw diagrams, and check that ou have enough information. It ma even be that ou have too much information and need to decide what is relevant and what is not. Making a plan can involve several different strategies. Your first idea ma not give ou a solution directl, but might suggest a better approach. You could look for patterns, draw more diagrams, or set up equations. You could also tr to solve a similar, simpler version of the problem and see if that gives ou some more insight. If ou have worked hard on the first two stages, carring out the plan should be the most straightforward stage. Work out our solution according to the plan and test it. You ma have to adjust the plan, but in doing so it will have given ou further insights that ma lead to a correct solution net time. It is important to look back at our solution and reflect on what ou have learned and wh this plan worked when others did not. This will help ou to become more confident in solving the net problem and the one after that. In this chapter and the net, ou will learn how to solve problems using mathematical models that involve particular tpes of function, namel linear, quadratic, polnomial and eponential functions. I t will be easier to stud this chapter if ou have alread completed Chapters, 14 and 17. In this chapter ou will learn: about linear functions and their graphs how to use linear models about quadratic functions and their graphs (parabolae) about the properties of a parabola: its smmetr, verte and intercepts on the -ais and -ais how to use quadratic models. 18.1 Linear models In Chapter, the general equation of a straight line (known as a linear equation) was given as = m + c. After meeting function notation in Chapter 17, ou now know that the same relationship can be written as f() = m + c, to epress that the input and the output, or f(), are related through a linear function whose graph is a straight line with gradient m and -intercept c. 18 Linear and quadratic models 51
Man practical situations ma be modelled with linear functions. For eample, monthl telephone costs tpicall include a fied charge plus a charge per minute; a plumber s fee is usuall made up of a fied call-out charge plus further costs depending on how long the job takes. In Chapter 4 ou studied currenc conversions, where each currenc is linked to another via a linear function. When ou are travelling abroad and using a different currenc on the metro or in shops and restaurants, a straight-line graph can help ou to quickl get an idea of prices. Imagine that Logan travels from Australia to India on holida. He changes Australian dollars (AUD) into Indian rupees (INR), and the echange rate at the time gives him 48 INR for 1 AUD. Using this echange rate, he draws a conversion graph that plots the number of rupees against the number of Australian dollars: 5 AUD 4 3 1 5 1 15 5 INR Now that Logan has this picture in his head, he can convert between INR and AUD with confidence. The graph quickl tells him, for instance, that: 1 INR = 5 AUD 4 AUD 19 INR. The gradient of the graph is m = 1 and, as INR = AUD, the -ais 48 intercept is at the origin. So the function can be written as A() = 1 48, where represents the number of Indian rupees and A() the number of Australian dollars. Logan takes a tai. The tai compan charges a fied fee of 16 INR for all journes, plus 1 INR for each kilometre travelled. The total cost of a journe is also a linear function. 5 Topic 6 Mathematical models
If Logan travels kilometres, he pas 1 INR for this distance on top of the fied charge of 16 INR, so the total cost function is C() = 16 + 1. Suppose Logan travels 8 km. How much will he pa? C(8) = 16 + 8 1 = 96 INR If Logan pas 1 INR for a tai ride, how far has he travelled? 1 = 16 + 1 1 16 = 1 = 1.4 km He visits a friend, and the friend recommends another tai compan, which has a lower initial charge but a higher charge per kilometre. This compan s cost function is D() = 4 + 14. Logan needs to make a km trip. He wants to know which of the two companies will be cheaper; he also wants to find out for what distance both companies will charge the same amount. A graph can answer both questions. Drawing the two linear functions on a GDC makes it simple to find the break-even point where the charges of both companies will be the same it is the point of intersection of the two lines: TEXAS CASIO For = : C() = 16 + 1 = 16 INR D() = 4 + 14 = 84 INR so the first compan will be cheaper. Looking at the graph, the break-even point occurs at 3 km, for which the cost is 46 rupees. For journes longer than 3 km Logan should use the first compan, but for short journes the second compan would be cheaper. 18 Linear and quadratic models 53
Worked eample 18.1 Q. Nimmi and her Theatre Studies group are presenting a show at the Iowa Festival Fringe. Nimmi is in charge of publicit and needs to order leaflets to advertise their show. A local printer quotes her $4 to set up and cents for ever leaflet. (a) Write down the cost function in the form C(p) = mp + c, where p is the number of leaflets and C(p) is total cost of printing them. (b) What is the cost of 6 leaflets? She finds a printer on the Internet who will charge her $3 as an initial cost, then 3 cents for ever leaflet. (c) Write down the cost function for the second printer in the form D(p) = mp + c. (d) If the bill from the online printer is $77, how man leaflets did Nimmi order? First, make sure ou are working with consistent units. There are 1 cents in a dollar, so cents = $.. (e) What is the break-even point where the two printers charge the same? A. (a) C(p) = 4 +.p To fi nd the cost of 6 leafl ets, substitute 6 for p in the function s formula. (b) C(6) = 4 + 6. = 5 so the cost is $5. This is similar to part (a), but with different values of m and c. (c) D(p) = 3 +.3p The bill from the online printer is D(p), so we need to solve D(p) = 77. (d) 77 = 3 +.3p 45 =.3p p = 15 54 Topic 6 Mathematical models
The break-even point is where C(p) = D(p). Plot both functions on our GDC and look for their intersection (see.1 (a) Solving linear equations using a graph on page 65 of the GDC chapter if ou need a reminder of how to use our GDC here). continued... (e) TEXAS The break-even point is at 8 leaflets. CASIO Worked eample 18. Q. A compan that makes hinges has fied costs of 1,. Each hinge costs 5 to make and is sold for 1.5. (a) Write down the cost function. (b) Write down the revenue function. (c) Find the point at which the compan starts to make a profit. The cost function is made up of the fi ed cost plus the cost of making the hinges at 5 per hinge. (d) What is the profit when the compan makes 4 hinges? A. (a) Let H be the number of hinges made; then the will cost 5H in total to make. C(H ) = 1 + 5H The revenue is the amount earned from selling the hinges. (b) R(H ) = 1.5H The compan will start to make a profi t when the revenue fi rst becomes greater than the cost, so we need to look for the point where C(H) = R(H), i.e. the break-even point. You can solve this equation with algebra or b plotting graphs on our GDC. (c) TEXAS CASIO 18 Linear and quadratic models 55
The profi t is the revenue minus the cost. To fi nd the profi t from 4 hinges, substitute 4 for p in the profi t function. continued... 1 + 5H = 1.5H 1 = 7.5H H = 1333.33... The compan starts to make a profit when it manufactures more than 1333 hinges. (d) P(H) = R(H) C(H) = 1.5H (1 + 5H) P(4) = 3 = 8 so the profit is 8. Eercise 18.1 1. The Thompsons are planning a holida. The want to rent a famil car and see the following two advertisements in a newspaper: ZOOM CAR RENTALS Flat fee: $3 plus $. per mile SAFE RIDE RENTALS Flat fee: $5 plus $.15 per mile The cost of renting a car from Zoom can be epressed as: C = 3 +.m where C is the total cost in dollars and m is the number of miles driven. (a) Write a similar equation for the cost of renting a car from Safe Ride. (b) What is the total number of miles that would make rental costs the same for both companies? (c) Which of the two companies should the Thompsons use if the plan to drive at least 1 miles? Justif our answer.. The Martins have switched their natural gas suppl to a compan called De GAS. The monthl bill consists of a standing charge of 6.5 and an additional charge of.35 per unit of gas used. (a) If the Martins use n units of gas in a month, write an equation to represent the total monthl charge, C(n), in pounds. (b) Calculate their bill when the Martins use 13 units in a calendar month. (c) The Martins were charged 7.9 in one month. Calculate the number of units of gas the used that month. 56 Topic 6 Mathematical models
3. The Browns are reviewing their electricit consumption. Their monthl bill includes a standing charge of 5.18 and an additional charge of.13 per kwh of electricit used. (a) Assuming the Browns use n units (kwh) of energ per month, write an equation to represent the total monthl cost of electricit, C(n), in pounds. (b) In one month the Browns used 5 units of electricit. Calculate the total cost of the energ used. (c) The electricit bill for the Browns was 89.68 in one calendar month. Determine the number of units used in that month. 4. Emma is considering the following two advertisements in the local newspaper: GYM BUDDIES $1 annual subscription plus $5 per visit FIT MATES $3 annual subscription plus $ per visit (a) For how man visits in a ear will the cost of the two gm services be the same? (b) Which of the two gms should Emma use if she plans to make: (i) no more than 5 visits a ear? (ii) at least 7 visits a ear? (c) Work out the difference in cost for (i) and (ii) in part (b). 5. Larisa and her friends are setting up a Young Enterprise compan in their school to design and sell birthda cards. Initial set-up costs amount to 1. The cost of producing each card is 9 pence. (a) Work out how man cards the have to sell to break even if each card is sold for: (i) 1.7 (ii) 1.9 (iii).1 (b) How much profit can the epect to make if the sell 3 cards at 1.9 each? (c) How much profit would be made if the sold the first 1 cards at 1.7 each and the net cards at.1 each? (d) What would the estimated profit be if the sold the first cards at 1.7 each and the net 1 cards at.1 each? 18 Linear and quadratic models 57
18. Quadratic functions and their graphs In Chapter ou learned that a quadratic equation: is an equation of the general form a + b + c = where a can have no solution, one solution or two possible solutions. The concept of a quadratic function is broader. A quadratic function is a function having the general form: f() = a + b + c where a Rather than concentrating on solutions to the equation a + b + c =, as we did in Chapter, we will now look at other properties of the function f() = a + b + c, and discuss how to use such a function in mathematical modelling. Using our GDC, draw graphs of f() = a for different values of a. Tr both positive and negative values. (See.G Graphs on page 645 of the GDC chapter for a reminder of how to plot graphs if ou need to.) 6 5 f() = f() = 6 f() = 5 4 4 3 3 1 4 3 1 1 1 3 4 3 f() = 4 1 f() =.5 4 3 1 1 1 3 4 1 4 3 1 1 1 3 4 f() =.1 3 4 f() = 5 6 5 6 f() = 5 Compare our graphs. What do ou notice? You should see that: If a is positive, the function has a minimum point. If a is negative, the function has a maimum point. As the magnitude of a increases (whether it is positive or negative), the parabola becomes steeper. As the magnitude of a decreases, the parabola becomes shallower. 58 Topic 6 Mathematical models
You should also have observed that the graph is smmetrical: there is a line of smmetr (also called an ais of smmetr) that cuts the curve in half so that each side is the mirror image of the other. The line of smmetr runs through the maimum or minimum point. This turning point, where the curve changes direction, is called the verte of the parabola. 6 line of smmetr 4 f() = 4 4 6 4 verte (, 4) Equa tion of the line of smmetr Using our GDC, draw graphs of f() = a + b + c with different values of a and b. See if ou can find a link between the values of a and b that will give ou an equation for the line of smmetr. 1 f() = 5 + 4 5 6 4 4 6 8 f() = + 3 4 5 f() = 15 + 13 1 15 = 3 = 5 = 15 4 18 Linear and quadratic models 59
Function f() = 5 + 4 f() = + 3 4 f() = 15 + 13 Location of line of smmetr = 5 = 3 = 15 4 hint Remember that the coeffi cient is the number that multiplies a variable. In the eamples above, look at the coefficient of the term and the coefficient of the term. In each case, if ou take the coefficient of the term (e.g. 5), change its sign (e.g. 5), divide b the coefficient of the term (e.g. 1) and then divide again b, ou get the location of the line of smmetr (e.g. = 5 ). This gives us a general formula for the line of smmetr of a quadratic function. a =πr For the quadratic function f() = a + b + c, the equation of the line of smmetr is = b a. T he verte of a parabola As the verte of the parabola lies on the line of smmetr, ou can use the line of smmetr as the -coordinate to calculate the corresponding -coordinate of the verte. For eample, f() = 5 + 4 has line of smmetr = 5. Substituting this value of into the function, we get: f 5 5 = 5 5 9 + 4 = 4 5 So the verte is the point (, 9 ). 4 You could also find the verte and line of smmetr of a parabola using our GDC, but in that case ou would work the other wa round, locating first the verte and then the line of smmetr. See 18.1 Using a graph to find the verte and line of smmetr of a parabola on page 68 of the GDC chapter if ou need a reminder of how. For eample, to find the verte of f() = 4 + 6, draw the graph of = 4 + 6 on our GDC, and find the coordinates of the maimum point. TEXAS CASIO The maimum point is at (3, 13), so the equation of the line of smmetr is = 3. 53 Topic 6 Mathematical models
Worked eample 18.3 Q. Draw the function f() = 7 + 8 on our GDC. (a) Write down the coordinates of the verte. (b) Write down the equation of the line of smmetr. (c) Write down the range of the function. If ou need a reminder of what the range is, look back at section 17.1. (d) What is the range of the function f()? A. TEXAS CASIO Enter = 7 + 8 into our GDC and draw its graph. Use our GDC to fi nd the coordinates of the minimum point. See section 18.1 on page 68 of the GDC chapter if ou need to. (a) The verte is at (1.75, 1.875). The equation of the line of smmetr is given b the -coordinate of the verte. (b) The line of smmetr is = 1.75. Check using the formula = b a. = ( ) = 1.75 As the minimum is at (1.75, 1.875), the graph never has a -value less than 1.875, though it can etend arbitraril far upwards. (c) The range is f() 1.875. 18 Linear and quadratic models 531
continued... (d) Draw the graph of f() on our GDC b entering = ( 7 + 8). You can see that this is just the graph of f() refl ected in the -ais, so the minimum has now become a maimum. TEXAS CASIO f() 1.875 Eercise 18.A 1. Draw the following functions on our GDC. In each case: (i) Write down the coordinates of the verte. (ii) State the equation of the line of smmetr. (iii) Write down the range of the function. (a) f() = + 3 (b) f() = 7 + (c) f() = + 6 (d) g() = 9 4 + 3 (e) g() = ( 3)( + 5) (f) g() = 1 ( + 1)( ) 1 3 5 (g) h ( ) = + (h) f() = 5.3 + 4.9.7 4. B sketching the graph of each of the following quadratic functions, find the coordinates of the verte and hence: (i) state the maimum or minimum value of the function (ii) determine the equation of the line of smmetr (iii) state the range of the function. (a) f() = 1.44.1 (b) g() = 1 (5 ) (c) h() = 3( + 1.7) + 8 53 Topic 6 Mathematical models
Intercepts on the and aes The quadratic function f() = a + b + c will alwas have one -intercept it is the value of f() when =, which is just the number c. With -intercepts, there are three different situations that can arise. (a) (b) (c) 6 6 6 4 4 4 4 6 4 6 4 6 4 The curve cuts the -ais at two points. 4 The curve touches the -ais at one point. 4 The curve does not touch or cross the -ais. In (a), there are two places where the function crosses the -ais; so there are two -intercepts. If ou are using the graph to solve a quadratic equation, ou can sa that there are two solutions to the equation, or two real roots. In (b), the parabola just touches the -ais tangentiall at one point, so there is onl one solution to the equation a + b + c =. Tangentiall means that the curve onl touches the ais at a single point and does not cross the ais. You will learn about tangents in Chapter. GDC Some calculators use the word zero instead of solution or root. In (c), the curve does not meet the -ais at all, so there are no -intercepts and thus no solutions to the equation a + b + c =. After drawing the graph of a quadratic function f() = a + b + c on our GDC, ou can easil use it to locate the solutions of a + b + c = (i.e. the -intercepts); see 18. Using a graph to find the zeros (roots) of a quadratic equation on page 681 of the GDC chapter if ou need a reminder of how to do this. 18 Linear and quadratic models 533
Worked e ample 18.4 Q. Use our GDC to draw the following functions: f() = 3 + 1, g() = 5 +, h() = + 4 (a) Write down the -intercept of h(). (b) Which function(s) has/have two -intercepts? (c) Which function has no -intercept? (d) Consider the function j() = + 4 + 4. How man -intercepts does this function have? Write down the value(s) of at the intercept(s). A. TEXAS CASIO If all three graphs are plotted on the same aes, ou can tell them apart b observing that g() is the curve with a maimum point, and h() is the curve that goes through (, ). The -intercept is the point on the graph where =. The minimum point of f() is above the -ais, so the curve will never touch or cross the -ais. (a) The -intercept of h() is (, ). (b) g() and h() have two -intercepts. (c) f() has no -intercept. Draw the graph of j(). It touches the -ais at the point (, ). Alternativel, note that j() = h() + 4, so the graph of j() is just the graph of h() shifted up b 4 units; the verte (, 4) of h() becomes the verte (, ) of j() on the -ais. (d) TEXAS j() has one -intercept at =. CASIO 534 Topic 6 Mathematical models
Eercise 18.B 1. Find the coordinates of the points where the graph = 7 intersects the -ais.. Consider the following functions: f()= 6 13 5, g() = + 3 + 5, h() = 16 9 (a) Which of the three functions from above has: + 3 5 4 (i) two intercepts? (ii) no intercepts? (iii) one intercept? (b) State the -intercept of each of the three functions. 3. Draw the graphs of the following quadratic functions on our GDC, and in each case state the coordinates of: (i) the -intercepts (ii) the -intercept (a) = 3 1 (b) = 16 9 (c) = 8 ( 5) (d) = ( 3) (1 ) (e) = 1 + ( 7)( + 3) (f) f() = 3.144 1.7( 5) 4. Consider the function f() = 7 1.4, 1 5.5. (a) State the -intercepts. GDC If ou are taking values from a sketch on our GDC, look carefull at the scale used for the window. If ou set a scale of 1, ou can count along the aes to get the and values. However, if ou have used ZOOM FIT (TEXAS) or AUTO (CASIO), the GDC will set the scale, which is often diverse and complicated, with fractional scales, giving ou a ver different picture. (b) Write down the equation of the line of smmetr. (c) State the range of f(). 5. The diagram shows the graph of the function f() = 18 11. Q is the verte of the parabola; P, R and S are the intercepts with the aes. Q 4 R P S 1 5 5 4 (a) Use our GDC to find the coordinates of points P, Q, R and S. (b) State the range of the parabola. (c) Write down the equation of the line of smmetr. 18 Linear and quadratic models 535
18.3 Quadratic model s Look around ou. How man parabolas have ou encountered toda? Have ou walked past a fountain or travelled over a suspension bridge? A quadratic function can be used to model all of these shapes. Ask two friends to throw a ball to each other, and watch the path of the ball or, if ou can, take a video of the ball-throwing and pla it back frame b frame. The path of the ball looks like a parabola. Galileo Galilei (1564 164) was born in Pisa, Ital. He began to stud medicine at the Universit of Pisa when he was 17, but did not complete his studies there, leaving in order to concentrate on philosoph and mathematics. In 1589, the universit appointed him professor of mathematics, and a ear later he wrote his book De Moto on the stud of motion. At the time, with the development of guns and cannons, there was much interest in understanding the motion of projectiles. Galileo used a combination of eperiments and mathematics to investigate how horizontal and vertical forces determine the path of a projectile; he showed that the path of a projectile can be modelled b a quadratic equation. 536 Topic 6 Mathematical models
Quadratic functions can be used to solve problems in man reallife situations. Using a quadratic function to solve a problem is quite straightforward if ou are given the function. Worked eample 18.5 You learned about solving quadratic equations in Chapter. Q. The height of a ball that has been thrown up into the air can be modelled b the quadratic function H(t) = a + bt 4.9t where t is the time in seconds after the ball is thrown, a is the initial height of the ball in metres and b is the initial velocit of the ball. (a) Am is plaing with a tennis ball. When the ball is at a height of 1. m above the ground she hits it verticall upwards with an initial velocit of m s 1. What is the ball s height after seconds? (b) Catherine can hit the ball harder. She hits the ball verticall upwards with an initial velocit of 6 m s 1 from a height of 1.5 m. When the ball reaches a height of 35.9 m, how long has it been in the air? Substitute given values of a and b into the epression for H(t). A. (a) For Am, a = 1. and b = H(t) = 1. + t 4.9t To fi nd the height after seconds, substitute t = in the function H(t). H() = 1. + 4.9 = 5.6 m Find the function for Catherine in the same wa. (b) For Catherine, a = 1.5 and b = 6 H(t) = 1.5 + 6t 4.9t If the height is 35.9 m, then H(t) = 35.9, and we need to solve this equation for t. 1.5 + 6t 4.9t = 35.9 Use an method ou like, for eample the equation solver on our GDC; see.3 Solving quadratic equations on page 656 of the GDC chapter if ou From GDC: t =.79 seconds need a reminder. 18 Linear and quadratic models 537
Using a quadratic function to solve a problem is more complicated if ou have to find an epression for the function first. Worked eample 18.6 Q. The area of a rectangular vegetable garden is 5 m. The perimeter of the plot is 65 m. What are the dimensions of the garden? 5 m w m First, epress the given information using algebra. The perimeter is made up of two lengths plus two widths. The area of the garden is length times width. We now have a quadratic equation. Now we can fi nd w b solving this quadratic equation. First rearrange it into the general form. l m A. P = w + l = 65 w + l = 3.5 so l = 3.5 w A = l w = 5 (3.5 w)w = 5 3.5w w = 5 w 3.5w + 5 = Use our GDC to solve (see Section.3 Solving quadratic equations on page 656 if ou need to.) From GDC: w = 1.5 or w = Calculate the corresponding values of l. Notice that either of the two solutions for w will give the same overall set of dimensions. If w = 1.5, then l = 3.5 1.5 = If w =, then l = 3.5 = 1.5 The dimensions are 1.5 m m. There is another wa that ou can investigate quadratic functions to model parabolas that ou see in everda life. Your GDC can fit a curve to an coordinates that ou give it, and find the equation that fits our data the closest. It can also give ou the product moment correlation coefficient, so that ou know how accurate the equation is as a model for our data. See 18.3 Using the statistics menu to find an equation on page 68 of the GDC chapter for a reminder if ou need to. You learned about product moment correlation coefficients in Chapter 1. 538 Topic 6 Mathematical models
Worked eample 18.7 Q. Kitt sees an arch that she thinks is in the shape of a parabola. She takes a photograph, and then las a grid over the picture to read off some coordinates. She finds the following coordinates: 4 5 6.4 1..8 4. 7. Enter the coordinates into our GDC and draw a scatter graph. (See 1.1 Drawing a scatter diagram of bivariate data on page 67 of the GDC chapter if ou need a reminder of how to draw a scatter diagram on our GDC.) Use this table of coordinates to determine if Kitt is correct; is the arch is in the shape of a parabola? A. TEXAS CASIO The scatter graph confi rms Kitt s idea because it shows a positive correlation between the and coordinates; use our GDC to fi t a curve to the points and fi nd an equation of the curve. (See 18.3 Using the statistics menu to fi nd an equation on page 68 of the GDC if ou need to.) hint You will see that the two GDCs have given different results for the coeffi cients of the quadratic function, even though both curves seem to fi t quite well; this is because the are using different formulae to calculate the quadratic curve of best fi t. If ou use this curve-fi tting technique in a project, it might be an issue to discuss when ou are assessing the validit of our results. Equation of the curve: =..8 +.53 r =.99 ( d.p.) Equation of the curve: =.7.58 +.63 r =.98 ( d.p.) The value of r suggests there is a strong positive correlation between the - and -coordinates and the equation of the curve calculated b the GDC. The equation of the curve contains and is therefore a quadratic, which suggests that Kitt was correct in thinking the arch is in the shape of a parabola. 18 Linear and quadratic models 539
It has been said that quadratic equations underpin all modern science. Is this true for other fields of stud too? Think about economics, or architecture and the principles of design. Using the same method, ou could fit a quadratic function to the path of the ball thrown between two people that we mentioned at the beginning of this section. Eercise 18.3 1. Sabina hits a tennis ball verticall upwards from an initial height of 1.4 metres giving the ball an initial velocit of m s 1. The flight of the ball can be modelled with the equation: h(t) = 1.4 + t 4.9t where h(t) is the height of the ball above the ground, t seconds after Sabina hit it. (a) Find the height of the ball after 1.5 seconds. (b) Use our GDC to draw the graph of h(t) and hence find the maimum height of the ball above the ground during its flight. (c) Sabina catches the ball when it falls back to a height of 1.4 metres. Find the total time that the ball was in flight.. Joe reckons that the approimate distance it takes to stop a car, depending on the speed at which the car is travelling, can be modelled b the equation: =.555 + 1.11.6494 where is the stopping distance in feet and is the speed of the car in miles per hour. Use the model to predict estimates of the stopping distances in the following table to 1 d.p. Speed, (mph) 3 4 5 6 7 Stopping distance, (feet) 3. The profit made b a large Slovenian compan can be modelled as a function of the ependiture on advertising: P() = 4.8 + 3.6. where P is the profit in millions of euros and is the ependiture on advertising in millions of euros. (a) Use our GDC to sketch the graph of P() for 3. (b) Hence find the ependiture on advertising that will maimise the profit. (c) Calculate the estimated profit when the compan spends 5 million on advertising. 54 Topic 6 Mathematical models
4. Tom hits a golf ball. The height of the ball can be modelled b the equation: h(t) = 7t t where h(t) is the height of the ball in metres, t seconds after Tom hit it. (a) What was the height of the ball after 3 seconds? (b) After how long did the ball reach a height of 1 metres for the first time? (c) What was the maimum height of the ball? (d) For how long was the ball more than 8 metres above the ground? 5. In a game of cricket, the batsman hits the ball from ground level. The flight of the ball can be modelled b a parabola. The path of the ball is epressed as: = 1.73.49 where is the height of the ball in metres above the ground and is the horizontal distance in metres travelled b the ball. 5 Height of ball (m) 15 1 5 1 3 4 Horizontal distance of ball (m) (a) Use our GDC to find: (i) the maimum height reached b the ball (ii) the horizontal distance travelled b the ball before it falls to the ground. (b) Work out the height of the ball when: (i) = 1 m (ii) = m. (c) Find the horizontal distance of the ball from the batsman when the ball first reaches a height of 13 metres. 18 Linear and quadratic models 541
Summa r You should know: how to identif linear functions, f () = m + c, and their graphs how to use linear models for solving practical problems how to identif quadratic functions, f () = a + b + c, and their graphs (parabolae) about the various properties of a parabola: b the equation of its line of smmetr, = a how to find the coordinates of its verte how to find its intercepts on the -ais and -ais how to use quadratic models to solve practical problems. 54 Topic 6 Mathematical models
Mied eamination practice Eam-stle questions 1. Draw the graph of the following functions on our GDC. In each case: (i) Write down the coordinates of the verte. (ii) State the equation of the line of smmetr. (iii) Write down the range of the function. (a) = (5 9) (b) = 7 (c) = 3 8. Sketch the graph of f() = 11 4 6 on our GDC. (a) Find the maimum or minimum value of the function. (b) Determine the equation of the line of smmetr. (c) State the range of the function. 3. The Smiths are looking for a plumber. The find this advert in the local paper. Sdner Plumber $14 call out + $98 per hour The total charge, C, for plumbing services can be written as: C = a + b where is the number of hours taken to finish the work. (a) State the values of a and b. (b) How much will it cost the Smiths if it takes 5 hours to complete the work? (c) In his net call out, the plumber charged Mr Jones $86. How long did the plumber work on this job? 4. The area of a rectangular soccer pitch is 45 m. The perimeter of the pitch is 7 m. (a) Taking the length of the pitch to be m, find an epression for the width of the pitch in terms of. (b) Hence show that satisfies the equation 135 + 45 =. (c) Solve the equation to find the dimensions of the pitch. 5. The trajector of a golf ball is in the form of a parabola. The path of the ball can be modelled b the equation: =.57.375 where is the distance in metres from where the ball was hit and is the height of the ball in metres above the ground. (a) Find the maimum height reached b the ball. (b) How far does the ball travel before it hits the ground for the first time? (c) What was the horizontal distance of the ball from where it was hit when its height was 19 m? 18 Linear and quadratic models 543
6. The subscription to a mathematics revision website MATHSMANAGER is 5 per month. There is an additional charge of 6p per visit. The total monthl cost of using the site can be represented as: C = C + n t where: C is the total monthl cost C is the fied monthl subscription fee n is the number of visits in a month t is the charge per visit. (a) Calculate the total cost for: (i) Saif, who visited the site 11 times last month (ii) Jeevan, who made 4 visits to the site last month. A second website MATHS-PLUS-U charges a 3. monthl subscription and 65p per visit. (b) How much would it have cost Saif and Jeevan individuall if the had used MATHS-PLUS-U instead? (c) Jack intends to revise intensel over the two months preceding his mock eaminations. He is planning to visit one of the revision websites at least 1 times. (i) Which of the two sites will be cheaper for him to use? (ii) What is the difference in cost between using these sites? Past paper questions 1. The function Q(t) =.3t.65t + 5 represents the amount of energ in a batter after t minutes of use. (a) State the amount of energ held b the batter immediatel before it was used. (b) Calculate the amount of energ available after minutes. (c) Given that Q(1) = 19.5, find the average amount of energ produced per minute for the interval 1 t. (d) Calculate the number of minutes it takes for the energ to reach zero. [Total 6 marks] [Ma 6, Paper 1, TZ, Question 7] ( IB Organization 6) See also the past paper questions at the end of Chapter. 544 Topic 6 Mathematical models
Chapter 19 Eponential and polnomial functions Eponential growth is a phrase we hear ever da on the news and in general life. But what does eponential mean? And what is eponential growth? The graph below shows the increase (in millions) in number of users of the social network site Facebook over a five and a half ear period. It demonstrates the tpical shape of an eponential function: the increase in the vertical ais starts off ver small and gradual and then increases in massive jumps of size despite the changes in the horizontal ais maintaining the same size. The graph below demonstrates that Facebook has seen an eponential growth in its number of users from 4 to 1. Number of users (in millions) 6 5 4 3 Facebook user growth In this chapter ou will learn: about eponential functions and their graphs how to find horizontal asmptotes for eponential functions how to use eponential models about functions of the form f() = a n + b m +, where n and m are positive integers, and their graphs how to use models of the form f() = a n + b m + where n, m Z + how to use graphs to interpret and solve modelling problems. 1 Dec-4 Jun-5 Dec-5 Jun-5 Dec-6 Jun-7 Dec-7 Date Jun-8 Dec-8 Jun-9 Dec-9 Jun-1 An eponential function gets its name from the fact that the independent variable is an eponent in the equation and this causes the dependent variable to change in ver large jumps. It will be easier to stud this chapter if ou have alread completed Chapters, 14, 17 and 18. 19.1 Eponential functions and their graphs Living organisms that reproduce seuall have two biological parents, four biological grandparents and eight great-grandparents. So the number of forebears (ancestors) doubles with each generation. You can demonstrate this eample using a table: Generation 1 3 4 5 6 Number of forebears 4 8 16 3 64 Power of 1 3 4 5 6 19 Eponential and polnomial functions 545
You can show the same information using a graph: 15 Number of forebears 1 f() = 5 6 8 Generation You can see from the graph that the growth in the number of forebears is ver fast. Human genealogists use an estimate of four generations per centur, so if ou tr to trace our famil back ears, ou could be looking for 8 = 56 different people! Some organisms reproduce ver fast, such as bacteria; the can achieve a population of millions in a ver short time. The tpe of growth described b the table and graph above is called eponential growth, and it is based on the function f() = a where a is a number greater than 1. The general form of an eponential function is: f() = ka + c where a >, is a variable that can be positive or negative, and k and c are constants (which can be positive or negative). The eponential function is ver interesting to eplore using a computer graphing package or a GDC. Start with the simplest case and draw graphs of f() = a for different values of a > 1: 5 4 15 3 1 5 4 6 5 546 Topic 6 Mathematical models
Net, draw the f() = a (note the negative sign in the eponent) for different values of a > 1: 5 4 3 15 1 5 6 4 Note that this is the same as drawing graphs of f() = a for positive values of a < 1. Tr it: draw the graphs of a for a =.5,.333,.5. Can ou see wh this is the case? You can also draw a set of graphs of f() = ka and f() = ka for the same value of a but different values of k: 5 1 4 3 3 8 3 6 4 3 3 1 1 3 4 19 Eponential and polnomial functions 547
Now tr drawing the graphs of f() = a ± c with the same value of a but different values of c. 6 4 + 4 1 4 4 6 8 4 After these eplorations, ou should find that: The curve alwas has a similar shape. You learned about asmptotes in Chapter 17. The curve levels off to a particular value. The horizontal line at this -value is called a horizontal asmptote; the curve approaches this line ver closel but never actuall reaches it. The graph of f() = a or a passes through the point (, 1). The graph of f() = ka or ka passes through the point (, k). The constant ±c moves the curve of ka up or down b c units, giving the graph a horizontal asmptote at c or c. Worke d eample 19.1 GDC See 17.3 Finding the horizontal asmptote on page 68 of the GDC chapter if ou need a reminder of how to fi nd horizontal asmptotes with our GDC. Q. Draw the following functions on our GDC and find the equation of the asmptote for each function. A. (a) (a) f() = 1.5 (b) g() = 3 (c) h() = 3 + TEXAS CASIO The asmptote is = (the -ais). 548 Topic 6 Mathematical models
continued... (b) TEXAS CASIO (c) The asmptote is =. TEXAS CASIO The asmptote is = 3. Ev aluating eponential functions Recall from section 17. that function notation provides an efficient shorthand for substituting values into a function. For eample, to find the value of f() = 6 + 3 when =, ou write f() to mean replace b in the epression 6 + 3. This is called evaluating the function f() at =. You can evaluate an function at a given value using either algebra or our GDC. If ou use our GDC, be careful to enter brackets where needed, especiall with negative values. Let s look at how to evaluate f() = 6 + 3 at = with three different methods. Using algebra: Replace with. eam tip See Learning links 1B on page 4 for a reminder about negative indices if ou need to. Using algebra is sometimes quicker and more accurate than using a GDC. If ou give our answer as a fraction, the answer is eact, whereas a recurring decimal is not. 1 f () 3 1 6 + 6 + = 6 = 3 9 55 9 19 Eponential and polnomial functions 549
Using our GDC to evaluate directl: TEXAS CASIO Using a graph on our GDC (see 19. Solving unfamiliar equations on page 684 of the GDC chapter, if ou need a reminder). TEXAS CASIO If ou are drawing an eponential graph on a GDC, ou need to be particularl careful in our choice of scale for the aes. In this eample, f() 6 alwas, so there is no need for a negative scale on the -ais; however, ou will onl get a good view of the shape of the graph if ou allow a wide enough domain for the -ais. Check that the same methods give f( 1) = 9 and f( 3) = 33. Wor ked eample 19. Q. Let f() = 1, g() = 1 + 3 and h() = 1.5.5. Find: Let s calculate these using algebra fi rst. (a) f() (b) g(5) (c) h( 4) A. (a) f ( ) 1 1 1 1 5 1 = = 4 4 (b) g( ) 1 + 3 5 1 + 43 = 44. (c) h( ( 4 ) = 15. ) = 1. 5 =. 5= 4. 5 TEXAS CASIO Now check the answers using our GDC. 55 Topic 6 Mathematical models
Eercise 19.1 1. Draw the following functions on our GDC, and find the equation of the asmptote in each case: (a) f() = 3 (b) f() = 3 (c) f() = 3 + 5 (d) f() = 3 4 (e) f() = 3 4. Draw the following functions on our GDC for the given domain, and in each case state the coordinates of the -intercept and the range. (a) g() = 5 for 3 6 (b) g() = 5 3 for 4 4 (c) g() = 5 3 + for 5 (d) g() = 5 + 7 for 1 (e) g() = 5 4 for 9 3. Draw the following functions on our GDC. In each case state: (i) the equation of the asmptote (ii) the coordinates of the -intercept (iii) the range. (a) f() = for 4 (b) f() = 3 for 6 1 1 (c) f() = for 8 (d) f() = 3 for 3 (e) f() = 6 + 5 for 5 (f) f() = 6 4 for (g) f() = 5 for 6 (h) f() = 4 5 for 1 4. For each of the following functions calculate: (i) f() (iii) f( 1) (a) f() = 5 (ii) f(3) 1 (iv) f (b) f() = 5 (c) f() = 5 + 3 (d) f() = 5 7 (e) f() = 1.68 5 (f) f() = 3.64 1 + 4 (g) f() = 5.7 1 3 8 (h) f() = 4.376 (i) f() = 34 1.45 1.5 3.578 (j) f() = 9 1 19 Eponential and polnomial functions 551
19. Eponential models Temperature ( C) 19 18 17 16 15 14 13 1 11 1 9 8 7 6 5 4 3 1 Powered b AKTSAG. Tpical cooling curves for solids Centre Surface Surrounding 1 Time (da) Eponential curves can be used as efficient models for a wide range of scientific and economic phenomena, such as the spread of a new technolog or the wa a virus infects a population. Although ou often hear the term eponential used to describe an growth pattern that shows a rapid increase (or decrease), it is important to be aware that eponential models will not alwas fit data outside a certain range; for instance, realisticall populations do not increase forever without limit. So, while eponential models are ver convenient, to make the best use of them we also need to understand their limitations. Eponential functions can be used to model both growth and deca ; in the modelling contet these phenomena are interpreted ver broadl. You alread encountered a particular form of eponential function when ou were studing compound interest, inflation and depreciation. For eample, the formula for compound interest: FV = PV(1+ r) n. Look back at Chapter 4 to refresh our memor on compound interest, inflation and depreciation. Compare f() = ka with the formula FV = PV(1+ r) n. If ou replace PV b k, n b, and (1 + r) b a, then ou can see that the compound interest formula is just an eponential function. In this section we will look at some other, non-financial, models, such as cooling and the growth or deca of populations. In a laborator that studies bacteria, the researchers are interested in the pattern of growth or decline of the bacterial populations. Suppose Jin has 3 g of bacteria originall and has established that the colon will grow according to the function: W(t) = 3 1.8 t where W(t) represents the mass of the colon at time t, measured in hours. 55 Topic 6 Mathematical models
She draws a graph of the growth she epects to see in the first week. Plotting the mass of the bacteria colon against time, she gets the following graph: 1 W 1 8 Mass (grams) 6 4 4 8 1 Time (hours) 16 t What is the mass of bacteria that she can epect to find at the end of the first week? At the end of the week, t = 7 4 = 168 hours, so the mass should be: W(168) = 3 1.8 168 = 114 g At the end of the week, something contaminates Jin s sample and the bacteria start ding. There are 5 g left after five das, and Jin wants to find a function that will model the deca of the population. Let the new function for modelling the deca of the bacterial population be called H(t), which represents the mass of the colon t hours after the contamination. (Note that this t has a different meaning from the t in W(t).) Jin immediatel writes H(t) = 114 a t, so that H() = 114, and needs to find a value for a. She knows that H(5) = 5 or, in other words, 5 = 114 a 5 4. So: 5 = 114 a 1 5 a 1 = 114 a 1 =.4386 hint The 114 in the epression for H(t) comes from the mass of the colon at the end of the fi rst week, when the contamination occurred. One method to find the value of a from the above equation is to draw the graph of the left-hand side of the equation on the same aes as the graph of the right-hand side of the equation, and see where the intersect. So Jin would enter 1 = 1 and =.4386 into her GDC and plot their 19 Eponential and polnomial functions 553
graphs (see 19.1 Solving growth and deca problems on page 683 of the GDC chapter for a reminder of how to solve eponential equations using our GDC): TEXAS CASIO Alternativel, she could take the 1th root of.4396 to get the same answer: 1 a =. 4386 =. 993 Therefore, the population deca function is H(t) = 114.993 t. With this function, it is possible to answer questions such as how long will it take for the mass of the colon to decrease to the original 3 g? All we need to do is solve the equation H(t) = 3 for the value of t: H(t) = 3 114.993 t = 3 B drawing the graphs of 1 = 114.993 and = 3, we see that t = 19 hours. TEXAS CASIO Worked eample 19.3 Q. The population of Inverness was 1 in 1. The town council estimates that the population is growing b 3% each ear. The growth can be modelled b the function P(t) = 1 1.3 t, where t is the number of ears after 1. (a) Estimate the population of Inverness in. Give our answer to the nearest ten people. (b) In which ear will the population be double that in 1? 554 Topic 6 Mathematical models
is ten ears after 1, so take t = 1 and evaluate P(1). continued... A. (a) P(1) = 1 1.3 1 = 161 (3 s.f.) If the population doubles, that means P(t) = 1. So solve this equation for t. Draw the graphs of 1 = 1.3 and =. Find their point of intersection; see section 19.1 in the GDC chapter if ou need to (b) 1 = 1 1.3 t Dividing through b 1, = 1.3 t TEXAS CASIO Remember that t is not the fi nal answer; ou need to add this value to 1. t = 3.4, so the population will have doubled in 33. hint The number e is a mathematical constant that often appears in growth or deca functions. It is an irrational number, e =.7188, with an infi nite number of decimals that do not show an repeating pattern. The concept of e is not in the sllabus for this course, but ou ma meet it in other contets. Worked eample 19.4 Q. A cup of coffee is left to cool. It is estimated that its temperature decreases according to the model T(t) = 9e.11t, where e is a constant with value approimatel.718, and time t is measured in minutes. (a) What is the initial temperature of the coffee? (b) Calculate the temperature after 5 minutes. (c) How long will it take for the coffee to cool to a room temperature of 1 C? Initial means at t =, so the initial temperature is found b evaluating T(). A. (a) T() = 9.718 = 9 1 = 9 C Substitute t = 5 into the eponential function. (b) T(5) = 9.718.11 5 = 51.9 C 19 Eponential and polnomial functions 555
Now we want to fi nd the value of t for which T(t) = 1. In other words, we need to solve this equation for t. continued... (c) 9.718.11t = 1 TEXAS CASIO On our GDC, draw the graphs of 1 = 9.718 and = 1. Find the point of intersection. t = 13. minutes GDC On the GDC ou use as the variable, but when writing down our answer, remember to use the correct letter given in the question, which is t in Worked eample 19.4. Eercise 19. 1. The population of a town is growing at a constant rate of % per ear. The present population is 48. The population P after t ears can be modelled as: P = 48 1. t (a) Calculate the population of the town: (i) after 5 ears (ii) after 1 ears (iii) after 14.5 ears. (b) Find how long it will take for the population of the town to reach: (i) 6 (ii) 1. (c) Find how long it will take for the population of the town to: (i) double (ii) treble.. The population of a large cit was 5.5 million in the ear. Assume that the population of the cit can be modelled b the function: P(t) = 5.5 1.5 t where P(t) is the population in millions and t is the number of ears after. (a) Calculate an estimate of the cit s population in: (i) 1 (ii) 18. 556 Topic 6 Mathematical models
(b) In what ear does the population of the cit become double that in? The population of another cit can be modelled as: P(t) = 6.5 1.1 t where t is the number of ears after the ear. (c) In which ear will the two cities have the same population? 3. Ali has been prescribed medication b his doctor. He takes a mg dose of the drug. The amount of the drug in his bloodstream t hours after the initial dose is modelled b: C = C.875 t (a) State the value of C. (b) Find the amount of the drug in Ali s bloodstream after: (i) 4 hours (ii) 8 hours. (c) How long does it take for the amount of drug in the bloodstream to fall below 4 mg? 4. The population of bacteria in a culture is known to grow eponentiall. The growth can be modelled b the equation: N(t) = 4e.375t (where e =.7188) Here N(t) represents the population of the bacteria t hours after the culture was initiall monitored. (a) What was the population of the bacteria when monitoring started? (b) Calculate N(4). (c) Work out N(48). (d) Calculate N(7). (e) How long does it take for the population of the bacteria to eceed 1? (f) How long does it take for the population of the bacteria to treble? 5. A liquid is heated to 1 C and then left to cool. The temperature θ after t minutes can be modelled as: θ = 1 e.4t (where e =.7188) (a) Calculate the temperature of the liquid after: (i) minutes (ii) 5 minutes (iii) hours. 19 Eponential and polnomial functions 557
(b) How long does it take for the temperature of the liquid to drop to 8 C? (c) Work out the time taken for the temperature of the liquid to drop to 4 C. 19.3 Po lnomial functions The function f() = a n + b m +, where the powers of (the n and m in the epression) are positive integers, is called a polnomial. You have alread studied two tpes of polnomial: linear functions, where the highest power of is 1, and quadratic functions, where the highest power of is. Now ou will meet some others. The degree of the polnomial is the highest power (or eponent, or inde) that appears in the function s epression; so a linear function is of degree 1, and a quadratic function is of degree. The simplest polnomial of degree n is the function f() = n. Draw some graphs of f() = n for various values of n: f() = 4 6 4 f() = 4 3 1 1 3 4 f() = 3 f() = 5 4 You can see that: The curves are all centred at the origin. If the power of is even, the curve is smmetrical about the -ais. If the power of is odd, the curve has 18 rotational smmetr about (, ). 558 Topic 6 Mathematical models
As ou add more terms to the function, the graph becomes more interesting and develops more curves. 6 4 f() = 3 4 + + 5 3 1 1 3 4 f() = 4 3 3 + 3 4 Name of function General algebraic epression Degree of polnomial Maimum number of turning points Linear f() = a + b 1 Quadratic f() = a + b + c 1 Cubic f() = a 3 + b + c + d 3 Quartic f() = a 4 + b 3 + c + d + e 4 3 Quintic f() = a 5 + b 4 + c 3 + d + e + f 5 4 For each degree of polnomial, there is a maimum number of turning points that the graph can have. But some polnomials of that degree will have fewer turning points; for eample, not all polnomials of degree 4 will have 3 turning points. It is best to draw an given polnomial function on our GDC to get an idea of what shape to epect. Worked eample 19.5 Q. Look at the following functions and write down the degree in each case: (a) f() = 5 4 6 5 + 4 3 (b) g() = 3 + 3 1 (c) h() = 7 + (d) j() = 1 3 + The term with the highest power is 6 5. A. (a) The degree is 5. 19 Eponential and polnomial functions 559
This has a rational term 1 (which is a negative power of ). Remember that in a polnomial all eponents of should be positive integers. continued... (b) It is not a polnomial. The term with the highest power is 7; this is a linear function. (c) The degree is 1. The term with the highest power is 3. (d) The degree is 3. As polnomial functions become more complicated, the ma also have more intercepts on the -ais. Finding these intercepts is ver similar to finding the solutions (or roots) of a quadratic equation, and ou can use our GDC to do this. (See 19.3 Solving polnomial equations (a) using a graph and (b) using an equation solver on page 685 of the GDC chapter if ou need a reminder.) See section 19.3 Solving polnomial equations on page 685 of the GDC chapter if ou need to. Worked eample 19.6 Q. Sketch the following curves with our GDC. In each case write down the -intercept and use our GDC to find an -intercepts. (a) f() = 4 5 + (b) g() = 5 + 3 (c) h() = 4 + 3 6 A. (a) TEXAS CASIO For the -intercept, look at the graph or evaluate f(). If ou zoom in near the origin, ou should see that there are two -intercepts there; be careful not to miss one. -intercept: (, ) -intercepts: ( 1.67, ), (, ), (.3, ), (1.47, ) 56 Topic 6 Mathematical models
continued... (b) TEXAS CASIO -intercept: (, ) -intercepts: (.571, ), (.758, ), (.31, ) (c) TEXAS CASIO -intercept: (, 6) -intercepts: ( 1.89, ), (1.36, ) Remember that in a polnomial function the powers of should all be positive integers. If a function of the form f() = a n + b m + contains one or more powers of that are negative integers, then f() is not a polnomial. Instead, it is a rational function. For eample, f() = 5 + 4 contains a negative power of (because 4 = 4 1 ) and can be written as f ) = 5 3 + 4, which is a ratio of two polnomials. The graphs of such rational functions will have = (the -ais) as a vertical asmptote. Rational functions and their graphs were covered in section 17.3. Eercise 19.3 1. Sketch the following curves on our GDC. In each case: (i) Write down the -intercept. (ii) Find an -intercepts. (a) f() = 3 7 + 6 (b) f() = 3 + 5 + 3 (c) f() = 8 5 3 (d) f() = 6 7 18 5 3 (e) f() = 3 + 3 4 (f) f() = 3 + 5 6 (g) f() = 3 + 13 + 6 (h) g() = 4 6 8 (i) g() = 4 + 7 3 6 7 + 4 (j) g() = 5 9 3 + 9 19 Eponential and polnomial functions 561
You learned about the domain and range in Chapter 17.. Sketch the following curves on our GDC for the stated domains. In each case: (i) write down the coordinates of the intercepts with the aes (ii) state the range of the function. (a) f() = 1 for 4 5 (b) f() = 1 + 7 for 1 7 (c) f() = 8 + 3 for 4 6 (d) f() = 3 5 + 6 for 4 (e) g() = 3 + 7 3 6 4 for 1 3 (f) g() = 3 + 5 + 4 for 4 6 (g) f() = 1 1 3 1 4 7 3 + for 1 (h) f() = 4 3 + + 6 for 19.4 Modelling with polnomial functions Some pra ctical situations can be described fairl well b a polnomial model. As with other models, the first step is to collect data and plot it on a graph. Then tr to find a curve that fits the data and helps ou to stud the problem in more depth. The function represented b the curve can be used to estimate values that were not collected as part of the original data set. You can also work out when that function will reach certain values of interest. Worked eample 19.7 Q. Bram is w orking in a laborator, measuring the speed (in metres per second) at which a particle drops through a certain liquid. He draws a graph of velocit against time, and thinks that it will fit a cubic model. After a bit more work, he decides that the equation v(t) = 34.3t 3 5.t + 6.8t +.16 would describe the data well during the first second. (a) Use this equation to estimate the velocit of the particle when t =.5 seconds. Finding the velocit at t =.5 means to replace t b.5 in the equation for v(t), i.e. to evaluate v(.5). (b) Calculate the time at which the velocit of the particle is.6 m s 1. A. (a) v(.5) = 34.3(.5) 3 5.(.5) + 6.8(.5) +.16 = 1.34 m s 1 56 Topic 6 Mathematical models
We want to fi nd the value of t for which v(t) =.6. This involves solving a polnomial equation. continued... (b) 34.3t 3 5.t + 6.8t +.16 =.6 TEXAS CASIO You can use the equation solver on our GDC (or ou could plot graphs and fi nd the point of intersection). See 19.3 (b) Solving polnomial equations using an equation solver on page 685 of the GDC chapter if ou need to. The velocit of the particle is.6 m s 1 after.113 seconds. Worked eample 19.8 Q. Risha has been monitoring the level of water in a local reservoir over the past three months. She finds that the depth of the water can be modelled b the function: D(t) =.4t 3.87t + 4.9t + 39 where t is measured in weeks. The beginning of the stud corresponds to t =, so evaluate D(). (a) Find the depth of water in the reservoir when Risha begins her stud. (b) After how man weeks is the depth of water in the reservoir at 5 m? (c) During Risha s stud, the water level first rises and then falls. What is the maimum depth of water that Risha measures? A. (a) D() = 39 m We want to fi nd the value of t for which D(t) = 5. This involves solving a cubic equation, which ou can do b drawing graphs on our GDC and fi nding the points of intersection (or using the equation solver program). See 19.3 Solving polnomial equations on page 685 of the GDC chapter if ou need to. (b).4t 3.87t + 4.9t + 39 = 5 TEXAS From GDC: t = 1 weeks CASIO 19 Eponential and polnomial functions 563
Draw the graph of D(t) and find the coordinates of the maimum point on the curve. (See 18.1 Using a graph to fi nd the verte and line of smmetr of a parabola on page 68 of the GDC chapter for a reminder of how to fi nd the minimum and maimum points, if ou need to.) continued... (c) TEXAS CASIO The maimum depth is 46.6 m, which occurs 3.5 weeks after Risha starts her stud. Eercise 19.4 1. Jana has been studing trends in the echange rate between the US dollar and the euro over the 1 months in 11. She has suggested an approimate cubic model; the equation of the modelling function is: f() =.3 3.88 +.748 + 1.58 where f() is the number of US dollars per euro, months after 1 Januar 11. (a) Using this model, estimate the echange rate of USD to EUR on: (i) 1 April 11 (ii) 1 August 11 (iii) 1 November 11. (b) Use our GDC to sketch the graph of f(), and hence estimate the peak value of the echange rate over the 1-month period.. The total population of the world in billions between 195 and 1 can be modelled b the following function: p () =.557 +.3554 +.145.7 3 +.1316 4.3718 5 where p() is the mid-ear population of the world in billions, decades after 195. (a) The actual mid-ear populations are given in the following table. Use the model to calculate the estimated populations. Work out the percentage error in using the model to estimate the population of the world. Decade 197 199 1 Population (billions) Percentage error Actual 3.76618 5.7864 6.848933 Estimated from model 564 Topic 6 Mathematical models
(b) Use the model to estimate the projected mid-ear population of the world in: (i) (ii) 3 (iii) 4. 3. Marko has studied trends in the price of silver on the commodities market over a si-ear period. He has suggested a model for the price per ounce of silver, in US dollars, over the si ears since. According to Marko, the price of silver can be modelled b the function f() =.669 5 + 1.383 4 5.9871 3 + 16.817.471 + 15.78 where f() is the price in USD per ounce of silver and is the number of ears after 1 Januar. (a) Use Marko s model to complete the following table: Year 4 6 Price of silver on 1 Januar (b) Use Marko s model to predict the price of silver on 1 Januar 5. The actual price of silver on the commodities market on 1 Januar 4 was US$9.8 per ounce. (c) Calculate the percentage error in using Marko s model to determine the price of silver on 1 Januar 4. 4. Martha found the following diagram in an Economics journal. The diagram shows the price of gold per ounce, in US dollars, over a tenear period. However, the numbers on one of the aes are missing. Price of gold Price in USD per ounce 4 6 8 1 1 Number of ears after 1 Jan 19 Eponential and polnomial functions 565
After some effort, Martha managed to find an approimate function g() to fit the trend of prices. Her function g() is: g() =.7 5 +.57 4.7818 3 +.36 3.773 + 78 where g() is the price in USD per ounce of gold ears after 1 Januar. (a) Use Martha s model to complete the following table: Year 4 6 8 1 1 Price of gold on 1 Januar (b) Use Martha s model to estimate the price of gold on 1 Januar 11. (c) If the actual price of gold on 1 Januar 11 was US$853.6 per ounce, calculate the percentage error in using Martha s model to estimate the price on 1 Januar 11. Su mmar You should know: what an eponential function is, and how to recognise one and its graph: f () = ka + c or f () = ka + c (where a Q +, a 1, k ) Recall from Chapter 1 that Q denotes rational numbers ; the superscript + means all positive rational numbers. how to find the horizontal asmptote of an eponential function how to use eponential models what a polnomial function is, and how to recognise one and its graph: f() = a n + b m + (where the powers of are positive integers) how to use polnomial functions as mathematical models. 566 Topic 6 Mathematical models