Functions and Graphs TERMINOLOGY

Transcription

1 5 Functions and Graphs TERMINOLOGY Arc of a curve: Part or a section of a curve between two points Asmptote: A line towards which a curve approaches but never touches Cartesian coordinates: Named after Descartes. A sstem of locating points (, ) on a number plane. Point (, ) has Cartesian coordinates and Curve: Another word for arc. When a function consists of all values of on an interval, the graph of = f] g is called a curve = f] g Dependent variable: A variable is a smbol that can represent an value in a set of values. A dependent variable is a variable whose value depends on the value chosen for the independent variable Direct relationship: Occurs when one variable varies directl with another i.e. as one variable increases, so does the other or as one variable decreases so does the other Discrete: Separate values of a variable rather than a continuum. The values are distinct and unrelated Domain: The set of possible values of in a given domain for which a function is defined Even function: An even function has line smmetr (reflection) about the -ais, and f] -g = -f] g Function: For each value of the independent variable, there is eactl one value of, the dependent variable. A vertical line test can be used to determine if a relationship is a function Independent variable: A variable is independent if it ma be chosen freel within the domain of the function Odd function: An odd function has rotational smmetr about the origin (0, 0) and where f] -g = -f] g Ordered pair: A pair of variables, one independent and one dependent, that together make up a single point in the number plane, usuall written in the form (, ) Ordinates: The vertical or coordinates of a point are called ordinates Range: The set of real numbers that the dependent variable can take over the domain (sometimes called the image of the function) Vertical line test: A vertical line will onl cut the graph of a function in at most one point. If the vertical line cuts the graph in more than one point, it is not a function

2 Chapter 5 Functions and Graphs 05 INTRODUCTION FUNCTIONS AND THEIR GRAPHS are used in man areas, such as mathematics, science and economics. In this chapter ou will stud functions, function notation and how to sketch graphs. Some of these graphs will be studied in more detail in later chapters. DID YOU KNOW? The number plane is called the Cartesian plane after Rene Descartes ( ). He was known as one of the first modern mathematicians along with Pierre de Fermat (60 665). Descartes used the number plane to develop analtical geometr. He discovered that an equation with two unknown variables can be represented b a line. The points in the number plane can be called Cartesian coordinates. Descartes used letters at the beginning of the alphabet to stand for numbers that are known, and letters near the end of the alphabet for unknown numbers. This is wh we still use and so often! Do a search on Descartes to find out more details of his life and work. Descartes Functions Definition of a function Man eamples of functions eist both in mathematics and in real life. These occur when we compare two different quantities. These quantities are called variables since the var or take on different values according to some pattern. We put these two variables into a grouping called an ordered pair.

3 06 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. Ee colour Name Anne Jacquie Donna Hien Marco Russell Trang Colour Blue Brown Gre Brown Green Brown Brown Ordered pairs are (Anne, Blue), (Jacquie, Brown), (Donna, Gre), (Hien, Brown), (Marco, Green), (Russell, Brown) and (Trang, Brown).. = + 5 The ordered pairs are (, ), (, ), (, ) and (, 5).. A B C D E The ordered pairs are (A, ), (B, ), (C, ), (D, ) and (E, ). Notice that in all the eamples, there was onl one ordered pair for each variable. For eample, it would not make sense for Anne to have both blue and brown ees! (Although in rare cases some people have one ee that s a different colour from the other.) A relation is a set of ordered points (, ) where the variables and are related according to some rule. A function is a special tpe of relation. It is like a machine where for ever INPUT there is onl one OUTPUT. INPUT PROCESS OUTPUT The first variable (INPUT) is called the independent variable and the second (OUTPUT) the dependent variable. The process is a rule or pattern.

4 Chapter 5 Functions and Graphs 07 For eample, in = +, we can use an number for (the independent variable), sa =. When = = + = As this value of depends on the number we choose for, is called the dependent variable. A function is a relationship between two variables where for ever independent variable, there is onl one dependent variable. This means that for ever value, there is onl one value. While we often call the independent variable and the dependent variable, there are other pronumerals we could use. You will meet some of these in this course. Investigation When we graph functions in mathematics, the independent variable (usuall the -value) is on the horizontal ais while the dependent variable (usuall the -value) is on the vertical ais. In other areas, the dependent variable goes on the horizontal ais. Find out in which subjects this happens at school b surveing teachers or students in different subjects. Research different tpes of graphs on the Internet to find some eamples. Here is an eample of a relationship that is NOT a function. Can ou see the difference between this eample and the previous ones? A B C D E In this eample the ordered pairs are (A, ), (A, ), (B, ), (C, ), (D, ) and (E, ). Notice that A has two dependent variables, and. This means that it is NOT a function.

5 08 Maths In Focus Mathematics Etension Preliminar Course Here are two eamples of graphs on a number plane... There is a ver simple test to see if these graphs are functions. Notice that in the first eample, there are two values of when = 0. The -ais passes through both these points.

6 Chapter 5 Functions and Graphs 09 There are also other values that give two values around the curve. If we drew a vertical line anwhere along the curve, it would cross the curve in two places everwhere ecept one point. Can ou see where this is? In the second graph, a vertical line would onl ever cross the curve in one place. So when a vertical line cuts a graph in more than one place, it shows that it is not a function. If a vertical line cuts a graph onl once anwhere along the graph, the graph is a function. If a vertical line cuts a graph in more than one place anwhere along the graph, the graph is not a function.

7 0 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. Is this graph a function? You will learn how to sketch these graphs later in this chapter. A vertical line onl cuts the graph once. So the graph is a function.. Is this circle a function? A vertical line can cut the curve in more than one place. So the circle is not a function.

8 Chapter 5 Functions and Graphs. Does this set of ordered pairs represent a function? ^-, h, ^-, h, ^05, h, ^, h, ^, h For each value there is onl one value, so this set of ordered pairs is a function.. Is this a function? Although it looks like this is not a function, the open circle at = on the top line means that = is not included, while the closed circle on the bottom line means that = is included on this line. So a vertical line onl touches the graph once at =. The graph is a function.

9 Maths In Focus Mathematics Etension Preliminar Course 5. Eercises Which of these curves are functions? ^, h, ^, -h, ^, h, ^0, h 0. ^, h, ^, -h, ^7, h, ^0, h

10 Chapter 5 Functions and Graphs. Name Ben Paul Pierre Hamish Jacob Lee Pierre Lien Sport Tennis Football Tennis Football Football Badminton Football Badminton 5. A B C 7 D E 5 F 7 G Function notation If depends on what value we give in a function, then we can sa that is a function of. We can write this as = f] g. EXAMPLES. Find the value of when = in the equation = +. When = : = + = + =. If f] g = +, evaluate f (). f] g = + f ] g = + = Notice that these two eamples are asking for the same value and f () is the value of the function when =. If = f] g then f ( a ) is the value of at the point on the function where = a

11 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. If f ] g = + +, find f ]-g. This is the same as finding when = -. f (- ) = ]- g + ( - ) + = =-. If f ] g = -, find the value of f ]-g. f ( ) = - f (- ) = ]-g - ]-g =-- =-. Find the values of for which f ] g = 0, given that f ] g = Putting f () = 0 is different from finding f (0). Follow this eample carefull. i.e. f ( ) = = 0 ( + 5)( - ) = = 0, - = 0 =- 5 =. Find f ] g, f ] g, f ] 0g and f] -g if f] g is defined as + when $ f ] g = ) - when. Use f() = + when is or more, and use f() = - when is less than. f ( ) = ( ) + since $ = f ( ) = ( ) + since $ = 0 f ( 0) =-( 0) since 0 = 0 f (- ) = -(-) = 8 since - 5. Find the value of g] g+ g] -g- g] g if when g ] g = * - when - # # 5 when -

12 Chapter 5 Functions and Graphs 5 g ( ) = ( ) - since - # # = g (- ) = 5 since - - g ( ) = since = 9 So g( ) + g( -)- g( ) = =- DID YOU KNOW? Leonhard Euler (707 8), from Switzerland, studied functions and invented the term f() for function notation. He studied theolog, astronom, medicine, phsics and oriental languages as well as mathematics, and wrote more than 500 books and articles on mathematics. He found time between books to marr and have children, and even when he went blind he kept on having books published. 5. Eercises. Given f] g = +, find f ] g and f ]-g.. If h ] g = -, find h] 0g, h] g and h ]-g.. If f] g =-, find f] 5g, f] -g, f] g and f ]-g.. Find the value of f] 0g + f] -g if f] g = Find f ]-g if f] g = If f] g = - 5, find when f] g =. 7. Given f] g = +, find an values of for which f] g = If f] g =, find when f] g = Find values of z for which f] zg = 5 given f] zg = z If f] g = - 9, find f^ph and f] + hg.. Find g] - g when g ] g = If f] g = -, find f] kg as a product of factors.. Given f] tg = t + t +, find t when f] tg = 0. Also find an values of t for which f] tg = 9.. Given f] tg = t + t - 5, find the value of f] bg- f] -bg. for 5. f] g = ) for # Find f] 5g, f] g and ]-g. Z - if $ ] 6. f] g = [ + if - ] if # - \ Find the values of f] g- f] - g+ f] -g. We can use pronumerals other than f for functions.

13 6 Maths In Focus Mathematics Etension Preliminar Course 7. Find g] g + g] 0g + g] -g if when $ 0 g ] g = ) when 0 8. Find the value of f] g - f] g + f] -g when for f] g = * for - # # for - 9. Find the value of f] -g - f] g - for $ if f( ) = * + - for 0. If f] g - - = - (a) evaluate f () (b) eplain wh the function does not eist for = (c) b taking several values close to, find the value of that the function is moving towards as moves towards.. If f] g = 5 +, find f] + hg - f] g in its simplest form. f] + hg- f] g. Simplif where h f] g = +. If f] g = 5 -, find f] g - f] cg in its simplest form.. Find the value of f^k h if + 5 for $ 0 f] g = * for 0 Z 5. If ] when $ f] g = [ 5 when 0 ] - + when # 0 \ evaluate (a) f (0) (b) f] g - f] g (c) f^-n h Graphing Techniques You ma have previousl learned how to draw graphs b completing a table of values and then plotting points. In this course, ou will learn some other techniques that will allow ou to sketch graphs b showing their important features. Intercepts One of the most useful techniques is to find the - and -intercepts. Everwhere on the -ais, = 0 and everwhere on the -ais = 0. For -intercept, = 0 For -intercept, = 0

14 Chapter 5 Functions and Graphs 7 EXAMPLE Find the - and -intercepts of the function f] g = For -intercept: = 0 0 = = ] + 8g] - g + 8 = 0, - = 0 =- 8, = For -intercept: = 0 = ] 0g + 7] 0g- 8 =-8 This is the same as = You will use the intercepts to draw graphs in the net section in this chapter. Domain and range You have alread seen that the -coordinate is called the independent variable and the -coordinate is the dependent variable. The set of all real numbers for which a function is defined is called the domain. The set of real values for or f ( ) as varies is called the range (or image) of f. EXAMPLE Find the domain and range of f] g =. You can see the domain and range from the graph, which is the parabola =. CONTINUED

15 8 Maths In Focus Mathematics Etension Preliminar Course Notice that the parabola curves outwards graduall, and will take on an real value for. However, it is alwas on or above the -ais. Domain: {all real } Range: { : $ 0} You can also find the domain and range from the equation =. Notice that ou can substitute an value for and ou will find a value of. However, all the -values are positive or zero since squaring an number will give a positive answer (ecept zero). Odd and even functions When ou draw a graph, it can help to know some of its properties, for eample, whether it is increasing or decreasing on an interval or arc of the curve (part of the curve ling between two points). If a curve is increasing, as increases, so does, and the curve is moving upwards, looking from left to right. If a curve is decreasing, then as increases, decreases and the curve moves downwards from left to right.

16 Chapter 5 Functions and Graphs 9 EXAMPLES. State the domain over which each curve is increasing and decreasing. The left-hand side of the parabola is decreasing and the right side is increasing. So the curve is increasing for and the curve is decreasing when. The curve isn t increasing or decreasing at. We sa that it is stationar at that point. You will stud stationar points and further curve sketching in the HSC Course.. The left-hand side of the curve is increasing until it reaches the -ais (where = 0). It then turns around and decreases until and then increases again. So the curve is increasing for 0, and the curve is decreasing for. 0 Notice that the curve is stationar at = 0 and =.

17 0 Maths In Focus Mathematics Etension Preliminar Course As well as looking at where the curve is increasing and decreasing, we can see if the curve is smmetrical in some wa. You have alread seen that the parabola is smmetrical in earlier stages of mathematics and ou have learned how to find the ais of smmetr. Other tpes of graphs can also be smmetrical. Functions are even if the are smmetrical about the -ais. The have line smmetr (reflection) about the -ais. This is an even function: For even functions, f] g = f] -g for all values of. Functions are odd if the have point smmetr about the origin. A graph rotated 80 about the origin gives the original graph. This is an odd function: For odd functions, f] - g = -f] g for all values of in the domain.

18 Chapter 5 Functions and Graphs EXAMPLES. Show that f] g = + is an even function. f] - g = ]- g + = + = f] g ` f] g = + is an even function. Show that f] g = - is an odd function. f] - g = ]-g - ]-g =- + =-^ -h =-f] g ` f] g = - is an odd function Investigation Eplore the famil of graphs of f] g = n. For what values of n is the function even? For what values of n is the function odd? Which families of functions are still even or odd given k? Let k take on different values, both positive and negative. k is called a parameter. Some graphics calculators and computer programs use parameters to show how changing values of k change the shape of graphs.. f] g = k n n. f] g = + k. f] g = ] + kg n 5. Eercises. Find the - and -intercept of each function. (a) = - (b) = 0 (c) + - = 0 (d) f] g = + (e) f] g = - (f) p ] g = (g) = (h) p ] g = + 5

19 Maths In Focus Mathematics Etension Preliminar Course (i) = + ]! 0g (j) g ] g = 9 -. Show that f] g = f] -g where f] g = -. What tpe of function is it?. If f] g = +, find (a) f^ h (b) 6 f ( )@ (c) f] -g (d) Is it an even or odd function?. For the functions below, state (i) the domain over which the graph is increasing (ii) the domain over which the graph is decreasing (iii) whether the graph is odd, even or neither. (a). Show that g] g = is an even function. 5. Show that f ( ) is odd, where f] g =. 6. Show that f] g = - is an even function. 7. Show that f] g = - is an odd function. (b) 8. Prove that f] g = + is an even function and hence find f] g- f] -g. 9. Are these functions even, odd or neither? (a) = - (b) = - (c) f] g = - - (d) = + (e) f] g = - 5 (c) - 0. If n is a positive integer, for what values of n is the function f] g = n (a) even? (b) odd? n. Can the function f] g = + ever be (a) even? (b) odd?

20 Chapter 5 Functions and Graphs (d) (e) Investigation Use a graphics calculator or a computer with graphing software to sketch graphs and eplore what effect different constants have on each tpe of graph. If our calculator or computer does not have the abilit to use parameters (this ma be called dnamic graphing), simpl draw different graphs b choosing several values for k. Make sure ou include positive and negative numbers and fractions for k. Alternativel, ou ma sketch these b hand.. Sketch the families of graphs for these graphs with parameter k. (a) = k (b) = k (c) = k (d) = k k (e) = What effect does the parameter k have on these graphs? Could ou give a general comment about = k f] g?. Sketch the families of graphs for these graphs with parameter k. (a) = ] + kg (b) = + k (c) = + k (d) = + k (e) = + k What effect does the parameter k have on these graphs? Could ou give a general comment about = f] g + k? CONTINUED

21 Maths In Focus Mathematics Etension Preliminar Course. Sketch the families of graphs for these graphs with parameter k. (a) = + k (b) = ] + kg (c) = ] + kg (d) = ] + kg (e) = + k What effect does the parameter k have on these graphs? Could ou give a general comment about = f] + kg? When k 0, the graph moves to the left and when k 0, the graph moves to the right. For the famil of functions = k f] g, as k varies, the function changes its slope or steepness. For the famil of functions = f ] g + k, as k varies, the graph moves up or down (vertical translation). For the famil of functions = f] + kg, as k varies, the graph moves left or right (horizontal translation). Notice that the shape of most graphs is generall the same regardless of the parameter k. For eample, the parabola still has the same shape even though it ma be narrower or wider or upside down. This means that if ou know the shape of a graph b looking at its equation, ou can sketch it easil b using some of the graphing techniques in this chapter rather than a time-consuming table of values. It also helps ou to understand graphs more and makes it easier to find the domain and range. You have alread sketched some of these graphs in previous ears. Linear Function A linear function is a function whose graph is a straight line. Gradient form: = m + b has gradient m and -intercept b General form: a + b + c = 0 Investigation Are straight line graphs alwas functions? Can ou find an eample of a straight line that is not a function? Are there an odd or even straight lines? What are their equations?

22 Chapter 5 Functions and Graphs 5 Use a graphics calculator or a computer with dnamic graphing capabilit to eplore the effect of a parameter on a linear function, or choose different values of k (both positive and negative). Sketch the families of graphs for these graphs with parameter k. = k. = + k. = m + b where m and b are both parameters What effect do the parameters m and b have on these graphs? EXAMPLE Sketch the function f] g = - 5 and state its domain and range. This is a linear function. It could be written as = - 5. Find the intercepts For -intercept: = 0 0 = = 6 5 = For -intercept: = 0 = ] 0g- 5 = Notice that the line etends over the whole of the number plane, so that it covers all real numbers for both the domain and range. Domain: {all real } Range: {all real } Notice too, that ou can substitute an real number into the equation of the function for, and an real number is possible for. The linear function a + b + c = 0 has domain {all real } and range {all real } where a and b are non-zero Special lines Horizontal and vertical lines have special equations.

23 6 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. Sketch = on a number plane. What is its domain and range? can be an value and is alwas. Some of the points on the line will be (0, ), (, ) and (, ). This gives a horizontal line with -intercept Domain: " all real, Range: " : =,. Sketch =- on a number plane and state its domain and range. can be an value and is alwas -. Some of the points on the line will be ^-, 0h, ^-, h and ^-, h. This gives a vertical line with -intercept Domain: " : =-, Range: " all real,

24 Chapter 5 Functions and Graphs 7 = a is a vertical line with -intercept a Domain:! : = a+ Range: {all real } = b is a horizontal line with -intercept b Domain: {all real } Range: " : = b, 5. Eercises. Find the - and -intercepts of each function. (a) = - (b) f] g = + (c) + - = 0 (d) - + = 0 (e) = 0. Draw the graph of each straight line. (a) = (b) - = 0 (c) = 5 (d) + = 0 (e) f ] g = - (f) = + (g) f ] g = + (h) + = (i) - - = 0 (j) + - = 0. Find the domain and range of (a) = 0 (b) = (c) =- (d) - = 0 (e) - = 0. Which of these linear functions are even or odd? (a) = (b) = (c) = (d) =- (e) = 5. B sketching - - = 0 and + - = 0 on the same set of aes, find the point where the meet.

25 8 Maths In Focus Mathematics Etension Preliminar Course Quadratic Function The quadratic function gives the graph of a parabola. f ] g = a + b + c is the general equation of a parabola. If a 0 the parabola is concave upwards If a 0 the parabola is concave downwards The pronumeral a is called the coefficient of. Applications The parabola shape is used in man different applications as it has special properties that are ver useful. For eample if a light is placed inside the parabola at a special place (called the focus), then all light ras coming from this light and bouncing off the parabola shape will radiate out parallel to each other, giving a strong light. This is how car headlights work. Satellite dishes also use this propert of the parabola, as sound coming in to the dish will bounce back to the focus.

26 Chapter 5 Functions and Graphs 9 The lens in a camera and glasses are also parabola shaped. Some bridges look like the are shaped like a parabola, but the are often based on the catenar. Research the parabola and catenar on the Internet for further information. Investigation Is the parabola alwas a function? Can ou find an eample of a parabola that is not a function? Use a graphics calculator or a computer with dnamic graphing capabilit to eplore the effect of a parameter on a quadratic function, or choose different values of k (both positive and negative). Sketch the families of graphs for these graphs with parameter k.. = k. = + k. = ] + kg. = + k What effect does the parameter k have on these graphs? Which of these families are even functions? Are there an odd quadratic functions?

27 0 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. (a) Sketch the graph of = -, showing intercepts. (b) State the domain and range. (a) This is the graph of a parabola. Since a 0, it is concave upward For -intercept: = 0 0 = - =! = For -intercept: = 0 = 0-5 = (b) From the graph, the curve is moving outwards and will etend to all real values. The minimum value is -. Domain: " all real, Range: " : $ -,. Sketch f] g = ] + g. This is a quadratic function. We find the intercepts to see where the parabola will lie. Alternativel, ou ma know from our work on parameters that f] g = ] + ag will move the function f] g = horizontall a units to the left. So f] g = ] + g moves the parabola f] g = unit to the left. For -intercept: = 0 0 = ] + g + = 0 =- For -intercept: = 0 = ] 0 + g =

28 Chapter 5 Functions and Graphs For the quadratic function f] g = (a) Find the - and -intercepts (b) Find the minimum value of the function (c) State the domain and range (d) For what values of is the curve decreasing? (a) For -intercept: = 0 This means f] g = 0 0 = = ] + g] - g + = 0, - = 0 =-, = For -intercept: = 0 f ] 0g = ] 0g + ] 0g- 6 =-6 (b) Since a 0, the quadratic function has a minimum value. Since the parabola is smmetrical, this will lie halfwa between the -intercepts. Halfwa between =- and = : - + =- Minimum value is f c- m f c- m= c- m + c- m -6 = - -6 =-6 So the minimum value is - 6. You will learn more about this in Chapter 0. CONTINUED

29 Maths In Focus Mathematics Etension Preliminar Course (c) Sketching the quadratic function gives a concave upward parabola , -6 From the graph, notice that the parabola is graduall going outwards and will include all real values. Since the minimum value is - 6, all values are greater than this. Domain: " all real, Range: ' : $ -6 (d) The curve decreases down to the minimum point and then increases. So the curve is decreasing for all -.. (a) Find the - and -intercepts and the maimum value of the quadratic function f ] g = (b) Sketch the function and state the domain and range. (c) For what values of is the curve increasing? (a) For -intercept: = 0 So f] g = 0 0 = = 0 ] - 5g] + g = 0-5 = 0, + = 0 = 5, = - For -intercept: = 0 f ] 0g =- ] 0g + ] 0g + 5 = 5

30 Chapter 5 Functions and Graphs Since a 0, the quadratic function is concave downwards and has a maimum value halfwa between the -intercepts =- and = = f ] g =- ] g + ] g+ 5 = 9 So the maimum value is 9. (b) Sketching the quadratic function gives a concave downward parabola From the graph, the function can take on all real numbers for, but the maimum value for is 9. Domain: " all real, Range: " : # 9, (c) From the graph, the function is increasing on the left of the maimum point and decreasing on the right. So the function is increasing when. 5.5 Eercises. Find the - and -intercepts of each function. (a) = + (b) =- + (c) f] g = - (d) = - - (e) = Sketch (a) = + (b) =- + (c) f] g = - (d) = + (e) =- - (f) f] g = ] - g

31 Maths In Focus Mathematics Etension Preliminar Course (g) f] g = ] + g (h) = + - (i) = (j) f] g = For each parabola, find (i) the - and -intercepts (ii) the domain and range (a) = 7 + (b) f] g = + (c) = (d) = (e) f] tg = - t. Find the domain and range of (a) = - 5 (b) f] g = - 6 (c) f] g = - - (d) =- (e) f] g = ] -7g 5. Find the range of each function over the given domain. (a) = for 0 # # (b) =- + for - # # (c) f] g = - for - # # 5 (d) = + - for - # # (e) =- - + for 0 # # 6. Find the domain over which each function is (i) increasing (ii) decreasing (a) = (b) =- (c) f] g = - 9 (d) =- + (e) f] g = ] + 5g 7. Show that f] g =- is an even function. 8. State whether these functions are even or odd or neither. (a) = + (b) f] g = - (c) =- (d) f] g = - (e) f] g = + (f) = - (g) = - - (h) = (i) p ] g = ] + g (j) = ] - g Absolute Value Function You ma not have seen the graphs of absolute functions before. If ou are not sure about what the look like, ou can use a table of values or look at the definition of absolute value. EXAMPLES. Sketch f] g = - and state its domain and range. Method : Table of values When sketching an new graph for the first time, ou can use a table of values. A good selection of values is - # # but if these don t give enough information, ou can find other values.

32 Chapter 5 Functions and Graphs 5 e.g. When =- : = - - = - = This gives a v-shaped graph Method : Use the definition of absolute value = - = & when when $ 0 This gives straight line graphs: = - ] $ 0g 5 = CONTINUED

33 6 Maths In Focus Mathematics Etension Preliminar Course =- - ] 0g = Draw these on the same number plane and then disregard the dotted lines to get the graph shown in method. = = - Method : If ou know the shape of the absolute value functions, find the intercepts. For -intercept: = 0 So f] g = 0 0 = - = ` =! For -intercept: = 0 f ( 0) = 0 - =-

34 Chapter 5 Functions and Graphs 7 The graph is V -shaped, passing through these intercepts If ou alread know how to sketch the graph of =, translate the graph of = - down unit, giving it a -intercept of -. From the graph, notice that values can be an real number while the minimum value of is -. Domain: {all real } Range: { : $ - }. Sketch = +. Method : Use the definition of absolute value. = + = ' + when + 0 -( + ) when + $ 0 This gives straight lines: = + when + $ 0 $ - 5 = CONTINUED

35 8 Maths In Focus Mathematics Etension Preliminar Course =- ] + g when + 0 i.e. =- - when - 5 = Draw these on the same number plane and then disregard the dotted lines. 5 = + = There is onl one solution for the equation 0. + = Can ou see wh? Method : Find intercepts For -intercept: = 0 So f ] g = 0 0 = + 0 = + - = For -intercept: = 0 f (0) = 0 + =

36 Chapter 5 Functions and Graphs 9 The graph is V -shaped, passing through these intercepts If ou know how to sketch the graph of =, translate it places to the left for the graph of = +. Investigation Are graphs that involve absolute value alwas functions? Can ou find an eample of one that is not a function? Can ou find an odd or even functions involving absolute values? What are their equations? Use a graphics calculator or a computer with dnamic graphing capabilit to eplore the effect of a parameter on an absolute value function, or choose different values of k (both positive and negative). Sketch the families of graphs for these graphs with parameter k. f] g = k. f] g = + k. f] g = + k What effect does the parameter k have on these graphs? The equations and inequations involving absolute values that ou studied in Chapter can be solved graphicall.

37 0 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES Solve. - = Sketch = - and = on the same number plane. The solution of - = occurs at the intersection of the graphs, that is, =-,.. + = - Sketch = + and = - on the same number plane. The graph shows that there is onl one solution. Algebraicall, ou need to find the possible solutions and then check them. The solution is =.. + Sketch = + and = on the same number plane.

38 Chapter 5 Functions and Graphs The solution of + is where the graph = + is below the graph =, that is, Eercises. Find the - and -intercepts of each function. (a) = (b) f] g = + 7 (c) f] g = - (d) = 5 (e) f] g =- + (f) = + 6 (g) f] g = - (h) = 5 + (i) = 7 - (j) f] g = + 9. Sketch each graph on a number plane. (a) = (b) f] g = + (c) f] g = - (d) = (e) f] g =- (f) = + (g) f] g =- - (h) = - (i) = + (j) f] g = +. Find the domain and range of each function. (a) = - (b) f] g = - 8 (c) f] g = + 5 (d) = - (e) f] g =- -. Find the domain over which each function is (i) increasing (ii) decreasing (a) = - (b) f] g = + (c) f] g = - (d) = - (e) f] g =- 5. For each domain, find the range of each function. (a) = for - # # (b) f] g =- - for - # # (c) f] g = + for - 7 # # (d) = - 5 for - # # (e) f] g =- for - # # 6. For what values of is each function increasing? (a) = + (b) f] g =- + (c) f] g = - 9 (d) = - - (e) f] g =- +

39 Maths In Focus Mathematics Etension Preliminar Course 7. Solve graphicall (a) = (b) (c) # (d) + = (e) - = 0 (f) - = (g) - (h) + # (i) - (j) - $ (k) + # 5 (l) - $ (m) - = + (n) - = - (o) - = + (p) + = + (q) + = - (r) - 5 = - (s) - = (t) - = + The Hperbola a A hperbola is a function with its equation in the form = a or =. EXAMPLE Sketch =. = is a discontinuous curve since the function is undefined at = 0. Drawing up a table of values gives: Class Discussion What happens to the graph as becomes closer to 0? What happens as becomes ver large in both positive and negative directions? The value of is never 0. Wh?

40 Chapter 5 Functions and Graphs To sketch the graph of a more general hperbola, we can use the domain and range to help find the asmptotes (lines towards which the curve approaches but never touches). The hperbola is an eample of a discontinuous graph, since it has a gap in it and is in two separate parts. Investigation Is the hperbola alwas a function? Can ou find an eample of a hperbola that is not a function? Are there an families of odd or even hperbolas? What are their equations? Use a graphics calculator or a computer with dnamic graphing capabilit to eplore the effect of a parameter on a hperbola, or choose different values of k (both positive and negative). Sketch the families of graphs for these graphs with parameter k k. =. = + k. = + k What effect does the parameter k have on these graphs? EXAMPLES. (a) Find the domain and range of f] g =. - (b) Hence sketch the graph of the function. This is the equation of a hperbola. To find the domain, we notice that -! 0. So! Also cannot be zero (see eample on page ). Domain: {all real :! } Range: {all real :! 0} The lines = and = 0 (the -ais) are called asmptotes. The denominator cannot be zero. CONTINUED

41 Maths In Focus Mathematics Etension Preliminar Course To make the graph more accurate we can find another point or two. The easiest one to find is the -intercept. For -intercept, = 0 = 0 - = = 0 Notice that this graph is a translation of = three units to the right = Asmptotes. Sketch =-. + This is the equation of a hperbola. The negative sign turns the hperbola around so that it will be in the opposite quadrants. If ou are not sure where it will be, ou can find two or three points on the curve. To find the domain, we notice that +! 0.! -! - For the range, can never be zero. Domain: {all real :! - } Range: {all real :! 0} So there are asmptotes at =- and = 0 (the -ais). To make the graph more accurate we can find the -intercept. For -intercept, = 0 =- 0 ( ) + =-

42 Chapter 5 Functions and Graphs The function f] g a = is a hperbola with b + c c domain & all real :! - 0 and b range {all real :! 0} 5.7 Eercises. For each graph (i) State the domain and range. (ii) Find the -intercept if it eists. (iii) Sketch the graph. (a) = (b) =- (c) f] g = + (d) f] g = - (e) = + 6 (f) f] g =- - (g) f] g = - (h) =- + (i) f] g = 6-6 (j) =- +. Show that f] g = is an odd function.. Find the range of each function over the given domain. (a) f] g = for - # # + 5 (b) = for - # # 0 + (c) f] g 5 = for - # # -

43 6 Maths In Focus Mathematics Etension Preliminar Course (d) f] g =- for - # # - (e) =- + for 0 # # 5. Find the domain of each function over the given range. (a) = for # # (b) =- for - # # - (c) f] g = for - # # (d) f] g =- for + - # # - 6 (e) = - for # # 6 Circles and Semi-circles The circle is used in man applications, including building and design. Circle gate A graph whose equation is in the form + a + + b + c = 0 has the shape of a circle. There is a special case of this formula: The graph of + = r is a circle, centre ^0, 0h and radius r Proof (, ) r

44 Chapter 5 Functions and Graphs 7 Given the circle with centre (0, 0) and radius r : Let (, ) be a general point on the circle, with distances from the origin on the -ais and on the -ais as shown. B Pthagoras theorem: c = a + b ` r = + EXAMPLE (a) Sketch the graph of + =. Is it a function? (b) State its domain and range. (a) This is a circle with radius and centre (0, 0). The radius is. - - The circle is not a function since a vertical line will cut it in more than one place. - - CONTINUED

45 8 Maths In Focus Mathematics Etension Preliminar Course (b) Notice that the -values for this graph lie between - and and the -values also lie between - and. Domain: { : - # # } Range: { : - # # } The circle + = r has domain:! : -r # # r+ and range: " : -r # # r, We can use Pthagoras theorem to find the equation of a more general circle. The equation of a circle, centre ( a, b ) and radius r is ] ag + ^ bh = r Proof Take a general point on the circle, (, ) and draw a right-angled triangle as shown. r (, ) - b b (a, b) a - a Notice that the small sides of the triangle are a and b and the hpotenuse is r, the radius. B Pthagoras theorem: c = a + b r = ] ag + ^ bh

46 Chapter 5 Functions and Graphs 9 EXAMPLES. (a) Sketch the graph of + = 8. (b) State its domain and range. (a) The equation is in the form + = r. This is a circle, centre (0, 0) and radius (b) From the graph, we can see all the values that are possible for and for the circle. Domain: { : - 9# # 9} Range: { : - 9# # 9}. (a) Sketch the circle ] g + ^ + h =. (b) State its domain and range. (a) The equation is in the form ] ag + ^ bh = r. ] g + ^ + h = ] g + _ ]- gi = So a =, b = - and r = CONTINUED

47 50 Maths In Focus Mathematics Etension Preliminar Course This is a circle with centre ^, -h and radius. To draw the circle, plot the centre point ^, -h and count units up, down, left and right to find points on the circle (, -) - -5 (b) From the graph, we can see all the values that are possible for and for the circle. Domain: { : - # # } Range: { : - # # 0}. Find the equation of a circle with radius and centre ^-, h in epanded form. You ma need to revise this in Chapter. This is a general circle with equation ] ag + ^ bh = r where a =-, b = and r =. Substituting: ] ag + ^ bh = r ] - ]- gg + ^ h = ] + g + ^ h = 9 Remove the grouping smbols. ] a + bg = a + ab + b So ] + g = + ] g] g + = + + ] a bg = a - ab + b So ^ h = - ^ h] g + = - + The equation of the circle is: = = = = 0

48 Chapter 5 Functions and Graphs 5 Investigation The circle is not a function. Could ou break the circle up into two functions? Change the subject of this equation to. What do ou notice when ou change the subject to? Do ou get two functions? What are their domains and ranges? If ou have a graphics calculator, how could ou draw the graph of a circle? B rearranging the equation of a circle, we can also find the equations of semi-circles. The equation of the semi-circle above the -ais with centre (0, 0) and radius r is = r - The equation of the semi-circle below the -ais with centre (0, 0) and radius r is =- r - Proof + = r = r =! r - This gives two functions: = r - is the semi-circle above the -ais since its range is $ 0 for all values. r -r r The domain is { : - r # # r} and the range is { : 0 # # r}

49 5 Maths In Focus Mathematics Etension Preliminar Course =- r - is the semi-circle above the -ais since its range is # 0 for all values. -r r -r The domain is { : - r # # r} and the range is { : - r # # 0} EXAMPLES Sketch each function and state the domain and range.. f] g = 9 - This is in the form f] g = r - where r =. It is a semi-circle above the -ais with centre (0, 0) and radius. - Domain: { : - # # } Range: { :0 # # }

50 Chapter 5 Functions and Graphs 5. =- - This is in the form =- r - where r =. It is a semi-circle below the -ais with centre (0, 0) and radius. - - Domain: { : - # # } Range: { : - # # 0} 5.8 Eercises. For each of the following (i) sketch each graph (ii) state the domain and range. (a) + = 9 (b) = 0 (c) ] g + ^ h = (d) ] + g + = 9 (e) ] + g + ^ h =. For each semi-circle (i) state whether it is above or below the -ais (ii) sketch the function (iii) state the domain and range. (a) =- 5 - (b) = - (c) = 6 - (d) =- 6 - (e) = Find the length of the radius and the coordinates of the centre of each circle. (a) + = 00 (b) + = 5 (c) ] g + ^ 5h = 6 (d) ] 5g + ^ + 6h = 9 (e) + ^ h = 8

51 5 Maths In Focus Mathematics Etension Preliminar Course. Find the equation of each circle in epanded form (without grouping smbols). (a) Centre (0, 0) and radius (b) Centre (, ) and radius 5 (c) Centre ^-, 5h and radius (d) Centre (, ) and radius 6 (e) Centre ^-, h and radius 5 (f) Centre ^0, -h and radius (g) Centre (, ) and radius 7 (h) Centre ^-, -h and radius 9 (i) Centre ^-, 0h and radius 5 (j) Centre ^-, -7h and radius Other Graphs You will meet these graphs again in the HSC Course. There are man other different tpes of graphs. We will look at some of these graphs and eplore their domain and range. Eponential and logarithmic functions EXAMPLES. Sketch the graph of f] g = and state its domain and range. If ou do not know what this graph looks like, draw up a table of values. You ma need to revise the indices that ou studied in Chapter. e.g. When = 0: = c = When =-: - = = = If ou alread know what the shape of the graph is, ou can draw it just using or points to make it more accurate.

52 Chapter 5 Functions and Graphs 55 This is an eponential function with -intercept. We can find one other point. When = = = You learned about eponential graphs in earlier stages of maths. From the graph, can be an real value (the equation shows this as well since an value substituted into the equation will give a value for ). From the graph, is alwas positive, which can be confirmed b substituting different values of into the equation. Domain: " all real, Range: " : 0,. Sketch f ] g = log and state the domain and range. Use the LOG ke on our calculator to complete the table of values. Notice that ou can t find the log of 0 or a negative number # # # From the graph and b tring different values on the calculator, can be an real number while is alwas positive. Domain:! : 0+ Range: " all real,

53 56 Maths In Focus Mathematics Etension Preliminar Course The eponential function = a has domain {all real } and range { : 0} The logarithmic function = log a has domain! : 0+ and range {all real } Cubic function A cubic function has an equation where the highest power of is. EXAMPLE. Sketch the function f] g = + and state its domain and range. Draw up a table of values If ou alread know the shape of =, f( ) = + has the same shape as f () = but it is translated units up (this gives a -intercept of ) The function can have an real or value: Domain: " all real, Range: " all real,

54 Chapter 5 Functions and Graphs 57 Domain and range Sometimes there is a restricted domain that affects the range of a function. EXAMPLE. Find the range of f] g = + over the given domain of - # #. The graph of f] g = + is the cubic function in the previous eample. From the graph, the range is {all real }. However, with a restricted domain of - # # we need to see where the endpoints of this function are. f ]- g = ]- g + =- + = f ] g = ] g + = 6 + = 66 Sketching the graph, we can see that the values of all lie between these points. (, 66) (-, ) Range: " : # # 66,

55 58 Maths In Focus Mathematics Etension Preliminar Course You ma not know what a function looks like on a graph, but ou can still find its domain and range b looking at its equation. When finding the domain, we look for values of that are impossible. For eample, with the hperbola ou have alread seen that the denominator of a fraction cannot be zero. For the range, we look for the results when different values of are substituted into the equation. For eample, will alwas give zero or a positive number. EXAMPLE Find the domain and range of f ] g = -. We can onl find the square root of a positive number or zero. So $ 0 $ When ou take the square root of a number, the answer is alwas positive (or zero). So $ 0 Domain:! : $ + Range: " : $ 0, 5.9 Eercises You ma like to simplif the function b dividing b.. Find the domain and range of (a) = + (b) f] g =- (c) = (d) f] g = (e) p ] g = (f) f] g = - - (g) + = 6 (h) f] tg = t - (i) g ( z) = + 5 z (j) f] g =. Find the domain and range of (a) = (b) = - (c) f] g = - (d) = - (e) f] g =- + 5 (f) = 5 - (g) = (h) =- 5 (i) f] g = + - (j) =. Find the -intercepts of (a) = ] - 5g (b) f] g = ] g] g] + g (c) = (d) g ] g = - 6 (e) + = 9

56 Chapter 5 Functions and Graphs 59. (a) Solve - $ 0. (b) Find the domain of f] g = Find the domain of (a) = - - (b) g] tg = t + 6t 6. Each of the graphs has a restricted domain. Find the range in each case. (a) = - in the domain - # # (b) = in the domain - # # (c) f ] g = in the domain - # # (d) = in the domain # # 5 (e) = in the domain 0 # # (f) = - in the domain - # # (g) =- in the domain - # # (h) = - in the domain - # # (i) = - - in the domain - # # (j) = in the domain 0 # # 7 7. (a) Find the domain for the function =. + (b) Eplain wh there is no - intercept for the function. (c) State the range of the function. 8. Given the function f] g = (a) find the domain of the function (b) find its range. 9. Draw each graph on a number plane (a) f] g = (b) =- (c) = - (d) p ] g = (e) g ] g = + (f) + = 00 (g) = + 0. (a) Find the domain and range of = -. (b) Sketch the graph of = -.. Sketch the graph of = 5.. For each function, state (i) its domain and range (ii) the domain over which the function is increasing (iii) the domain over which the function is decreasing. (a) = - 9 (b) f] g = - (c) = (d) f] g = (e) f] g =. (a) Solve - $ 0. (b) Find the domain and range of (i) = - (ii) =- -.

57 60 Maths In Focus Mathematics Etension Preliminar Course DID YOU KNOW? A lampshade can produce a hperbola where the light meets the flat wall. Can ou find an other shapes made b a light? Lamp casting its light Limits and Continuit Limits A line that a graph approaches but never touches is called an asmptote. The eponential function and the hperbola are eamples of functions that approach a limit. The curve = a approaches the -ais when approaches ver large negative numbers, but never touches it. That is, when " -, a " 0. Putting a - into inde form gives - a = a = Z 0 We sa that the limit of a as approaches - is 0. In smbols, we write lim a = 0. " - EXAMPLES + 5. Find lim. " 0 0 Substituting = 0 into the function gives, which is undefined. 0 Factorising and cancelling help us find the limit. + 5 ] + 5g lim = lim " 0 " 0 = lim ( + 5) " 0 = 5

58 Chapter 5 Functions and Graphs 6 -. Find lim. " - 0 Substituting = into the function gives, which is undefined lim = lim " - " ^ + h _ -i = lim " + = h+ h- 7h. Find lim. h " 0 h h+ h- 7h h^h + - 7h lim = lim h " 0 h h " 0 h = lim h h " 0 = - 7 Continuit Man functions are continuous. That is, the have a smooth, unbroken curve (or line). However, there are some discontinuous functions that have gaps in their graphs. The hperbola is an eample. If a curve is discontinuous at a certain point, we can use limits to find the value that the curve approaches at that point. EXAMPLES -. Find lim and hence describe the domain and range of the curve " - - =. Sketch the curve Substituting = into gives - 0 CONTINUED

59 6 Maths In Focus Mathematics Etension Preliminar Course Remember that!. - ] + g] - g lim = lim " = lim ( + ) " = - = is discontinuous at = since is undefined at that point. - This leaves a gap in the curve. The limit tells us that " as ", so the gap is at ^, h. Domain: " : all real,!, Range: " : all real,!, - = - ^ + h^ - h = - = + ` the graph is = + where! + -. Find lim and hence sketch the curve = + -. " Substituting =- into gives ^ - h^ + h lim = lim "- + "- ^ + h = lim ^ - h " - =- + - = is discontinuous at =- + ^ + h^ - h = + = - So the function is = - where! -. It is discontinuous at ^-, -h.

60 Chapter 5 Functions and Graphs Eercises. Find (a) lim + 5 " (b) lim t - 7 t " - (c) lim + - " + (d) lim " 0 h - h - (e) lim h " h (f) lim " 5-5 (g) lim "- (h) lim " - c - (i) lim c " c - - (j) lim " h + h - 7h (k) lim h " 0 h h - h + h (l) lim h " 0 h h - h + h - 5h (m) lim h " 0 h - c (n) lim " c - c. Determine which of these functions are discontinuous and find values for which the are discontinuous. (a) = - (b) = + (c) f ] g = - (d) = + (e) = -. Sketch these functions, showing an points of discontinuit. (a) = + + (b) = (c) = +

61 6 Maths In Focus Mathematics Etension Preliminar Course Further Graphs Graphs of functions with asmptotes can be difficult to sketch. It is important to find the limits as the function approaches the asmptotes. A special limit is lim = 0 " EXAMPLES. Find lim. " - + lim lim - + = - + = lim " - + = = " " (dividing b the highest power of ). Find (a) lim " + + (b) lim " (a) lim lim + + = " " + + = lim " = = 0

62 Chapter 5 Functions and Graphs 65 Since " 0 from the positive side when " +, we can write lim = + " (b) lim lim + + = + + = 0 "- "- Since " 0 from the negative side when " -, we can write lim = - - " + + is positive whether is + or -. Can ou see wh?. Find lim. " - Dividing b will give 0. Divide b. lim lim - = - = lim " - = - 0 = " " General graphs It is not alwas appropriate to sketch graphs, for eample, a hperbola or circle, from a table of values. B restricting the table of values, important features of a graph ma be overlooked. Other was of eploring the shape of a graph include: intercepts The -intercept occurs when = 0. The -intercept occurs when = 0. even and odd functions Even functions 6 f^- h = f( are smmetrical about the -ais. Odd functions 6 f^- h = -f( are smmetrical about the origin.

63 66 Maths In Focus Mathematics Etension Preliminar Course asmptotes Vertical asmptotes occur when f ] g! 0 and h ] g = 0, given g ] g f ] g =. h ] g Horizontal and other asmptotes are found (if the eist) when finding lim f] g. "! domain and range The domain is the set of all possible values for a function. The range is the set of all possible values for a function. EXAMPLES. Sketch =. - 9 Intercepts: For -intercept, = 0 0 = = 0 = So the -intercept is 0. For -intercept, = 0 0 = 0 - = 0 So the -intercept is 0 Tpe of function: ]-g f( - ) = ] - g - 9 = - 9 = f ( ) The function is even so it is smmetrical about the -ais. Vertical asmptotes: - 9! 0 ] + g] - g! 0 +! 0, -! 0! -,! So there are asmptotes at =!.

64 Chapter 5 Functions and Graphs 67 As " from LHS : ^ - h - f ( ) = ^ - h = - =- So " - As " from RHS : ^ + h + f ( ) = ^ + h = + =+ So " As " - from LHS : ^ - - h - f (- ) = ^ - - h = + =+ So " As " - from RHS : ^ + - h + f (- ) = ^ - + h = - =- So " - Horizontal asmptotes: lim lim - 9 = " " 9 - = lim " 9 - = - 0 = As " f ( ) = - 9 So as ", " from above You could substitute values close to on either side into the equation, sa.9 on LHS and. on RHS. You could substitute values close to - on either side into the equation, sa -. on LHS and - 9. on RHS. CONTINUED

65 68 Maths In Focus Mathematics Etension Preliminar Course You could substitute values such as 000 and to see what does as approaches!. As " - ]- g f (- ) = - ] g - 9 So as " -, " from above Domain: { : all real!! } Range: When, When -, # 0 When -, So the range is { :, # 0}. All this information put together gives the graph below.. Sketch f ( ) =. - Intercepts: For -intercept, = 0 0 = - 0 = 0 = So the -intercept is 0 For -intercept, = 0 0 = 0 - = 0 So the -intercept is 0. Tpe of function: ]-g f( - ) = ] - g - = - - =- +! -f ( ) The function is neither even nor odd.

66 Chapter 5 Functions and Graphs 69 Vertical asmptotes: -! 0! So there is an asmptote at =. As " from LHS : ^ - h - f ( ) = = - =- So " - As " from RHS : ^ + h + f ( ) = = + =+ So " You could substitute values close to on either side into the equation, sa.9 on LHS and. on RHS. e.g. When =. ]. g f (.) =. - =. Horizontal asmptotes: lim lim - = " " - = lim " - = - 0 = This means that as approaches!, the function approaches =. As " f ( ) = - So as ", " from above. As " - ]- g f (- ) = - - So as " -, " from above. Note: If we divide everthing b, we get. Divide b. 0 CONTINUED

67 70 Maths In Focus Mathematics Etension Preliminar Course This is not eas to see, so substitute values such as 000 and to see what does as approaches ±. e.g. When =- 000 ]-000g f (- 000) = =-998 The point ^-000, -998h is just above the line =. Domain: { : all real! } Range: When we find that an approimate range is 5 (substituting different values of ) When, # 0 So the range is { : 5, # 0} Putting all this information together gives the graph below. There is a method that combines all these features to make sketching easier. EXAMPLES. Sketch =. - 9 First find the critical points ( -intercepts and vertical asmptotes). = # ] + g] - g

68 Chapter 5 Functions and Graphs 7 -intercepts: = 0 0 = = 0 = asmptotes: ( + )( - ) = 0 =! These critical points, = 0,!, divide the number plane into four regions. Then sketch =, = + and = - on our graph. These are straight lines at the critical points. Look at the sign of the curve in each region. Region : = + = + + = - + ` # = ] + g] - g + # + = + # + =+ Region : = + = + + = - - ` # = ] + g] - g + # + = + # - + = - =- A graph is positive if it is above the -ais. The curve is above the -ais in this region. The curve is below the -ais in this region. CONTINUED

69 7 Maths In Focus Mathematics Etension Preliminar Course The curve is below the -ais in this region. The curve is above the -ais in this region. Check these! Region : = - = + + = - - ` # = ] + g] - g -#- = + # - + = - =- Region : = - = + - = - - ` # = ] + g] - g -#- = - # - + = + =+ Find an horizontal asmptotes. lim lim - 9 " " 9 - = from above lim " - fromabove - 9 = All this information put together gives the following graph.. Sketch = +. ] + g] - g Find the critical points. =- ( -intercept) =-0 = ^vertical asmptotesh

70 Chapter 5 Functions and Graphs 7 Use these to divide the number plane into regions and sketch = +, = + and = -. + Region : = ] + g] - g + = + # + =+ + Region : = ] + g] - g + = + # - =- For horizontal asmptotes + Region : = ] + g] - g - = + # - =+ + Region : = ] + g] - g - = - # - =- + + lim = lim " ] + g] -g " = lim " = lim = 0 - ] + g] -g " All this information put together gives the following graph. The -intercept is -.

71 7 Maths In Focus Mathematics Etension Preliminar Course Class Investigation You can eplore graphs of this tpe on a graphical calculator or b using computer software designed to draw graphs. 5. Eercises. Find (a) lim " (b) lim " + 5 (c) lim " + (d) lim " - (e) lim " (f) lim " (g) lim " (h) lim " (i) lim " + (j) lim " -. (a) Show that + + = (b) Find lim " + + (c) Find lim " -. Find (a) lim " + 7 (b) lim " 5. Sketch (a) 5 + = + (b) = - (c) = + (d) = + (e) = - (f) = + + (g) + = - - (h) = + (i) (j) = + = -. Find (a) lim " + 5 (b) lim " - + 5

72 Chapter 5 Functions and Graphs 75 Solving inequations graphicall There are different methods of solving inequations involving pronumerals in the denominator. You learned how to solve these using the number line in Chapter. EXAMPLES. Solve $ graphicall. + Sketch = and = on the same number plane. + The hperbola has domain {all real :! - } and range {all real :! 0}. For -intercept: = 0 = 0 + = = = + - The solution of $ occurs when the hperbola = is on or + + above the line =. CONTINUED

73 76 Maths In Focus Mathematics Etension Preliminar Course The two graphs intersect where + =. #( + ) = #( + ) + = ] + g = + - = - = - = The solution is - # - (from the graph).. Solve graphicall. - Sketch = and = on the same number plane. - The hperbola has domain {all real :! } and range {all real :! 0}. For -intercept: = 0 = 0 - =- = = - The solution of occurs when the hperbola = is below - - the line =.

74 Chapter 5 Functions and Graphs 77 The two graphs intersect where - = #( - ) = #( - ) - = - = The solution is, 5. Eercises Solve graphicall and algebraicall $ - $ - 6. # + 7. $ # - 0. $ + Regions Class Investigation How man solutions are there for $ +? How would ou record them all? Inequalities can be shown as regions in the Cartesian plane. You can shade regions on a number plane that involve either linear or non-linear graphs. This means that we can have regions bounded b a circle or a parabola, or an of the other graphs ou have drawn in this chapter. Regions can be bounded or unbounded. A bounded region means that the line or curve is included in the region.

75 78 Maths In Focus Mathematics Etension Preliminar Course EXAMPLE Sketch the region #. Remember that = is a vertical line with -intercept. # includes both = and in the region. Sketch = as an unbroken or filled in line, as it will be included in the region. Shade in all points where as shown = An unbounded region means that the line or curve is not included in the region. EXAMPLE Sketch the region -. - doesn t include =-. When this happens, it is an unbounded region and we draw the line =- as a broken line to show it is not included.

76 Chapter 5 Functions and Graphs 79 Sketch =- as a broken line and shade in all points where - as shown. Remember that =- is a horizontal line with -intercept -. 5 = For lines that are not horizontal or vertical, or for curves, we need to check a point to see if it lies in the region. EXAMPLES Find the region defined b. $ + First sketch = + as an unbroken line. On one side of the line, + and on the other side, +. To find which side gives +, test a point on one side of the line (not on the line). For eample, choose ^0, 0h and substitute into $ + 0 $ $ (false) CONTINUED

77 80 Maths In Focus Mathematics Etension Preliminar Course This means that ^0, 0h does not lie in the region $ +. The region is on the other side of the line. An point in the region will make the inequalit true. Test one to see this First sketch - = 6 as a broken line, as it is not included in the region. To find which side of the line gives - 6, test a point on one side of the line. For eample, choose ^0, h and substitute into ] g - ( ) 6-6 (true) This means that ^0, h lies in the region =6. + The equation + = is a circle, radius and centre ^0, 0h. Draw + = as a broken line, since the region does not include the curve.

78 Chapter 5 Functions and Graphs 8 Choose a point inside the circle, sa ^0, 0h (false) So the region lies outside the circle.. $ The equation = is a parabola. Sketch this as an unbroken line, as it is included in the region. Choose a point inside the parabola, sa ^, h. $ (true) So ^, h lies in the region. = Sometimes a region includes two or more inequalities. When this happens, sketch each region on the number plane, and the final region is where the overlap (intersect).

79 8 Maths In Focus Mathematics Etension Preliminar Course EXAMPLE Sketch the region #, - and #. Draw the three regions, either separatel or on the same set of aes, and see where the overlap.. Put the three regions together. If ou are given a region, ou should also be able to describe it algebraicall.

80 Chapter 5 Functions and Graphs 8 EXAMPLES Describe each region The shaded area is below and including = 6 so can be described as # 6. It is also to the left of, but not including the line =, which can be described as. The region is the intersection of these two regions: # 6 and. - - CONTINUED

81 8 Maths In Focus Mathematics Etension Preliminar Course The shaded area is the interior of the circle, centre (0, 0) and radius but it does not include the circle. The equation of the circle is + = or + =. You ma know (or guess) the inequalit for the inside of the circle. If ou are unsure, choose a point inside the circle and substitute into the equation e.g. (0, 0). LHS = + = = 0 ] RHSg So the region is Eercises. Shade the region defined b (a) # (b) (c) $ 0 (d) 5 (e) # + (f) $ - (g) + (h) (i) + - $ 0 (j) - 0. Write an inequation to describe each region. (a) (b) (c) = + -

82 Chapter 5 Functions and Graphs 85 (d) 5 = - 5. Shade the region (a) ] - g + # (b) ] - g + ^ - h # (c) ] + g + ^ - h (e) =. Shade each region described. (a) (b) + # 9 (c) + $ (d) # (e). Describe as an inequalit (a) the set of points that lie below the line = - (b) the set of points that lie inside the parabola = + (c) the interior of a circle with radius 7 and centre (0, 0) (d) the eterior of a circle with radius 9 and centre (0, 0) (e) the set of points that lie to the left of the line = 5 and above the line = 6. Shade the intersection of these regions. (a) #, $ - (b) $ -, - (c) #, $ - 5 (d) +, # - (e) #, + # 9 (f) -, + (g) #, $ (h) -, #, (i) # 0, + $ (j) - - # 7. Shade the region bounded b (a) the curve =, the -ais and the lines = and = (b) the curve =, the -ais and the lines = 0 and = (c) the curve + =, the -ais and the lines = 0 and = in the first quadrant (d) the curve =, the -ais and the lines = and = (e) the curve =, the + -ais and the lines = 0 and = 8. Shade the regions bounded b the intersection of (a), 5 and # (b), $ -, # - (c) # -, # +, - # 6 (d) $ -, #, + $ 9 (e), #, $ The first quadrant is where and values are both positive.

83 86 Maths In Focus Mathematics Etension Preliminar Course Application Regions are used in business applications to find optimum profit. Two (or more) equations are graphed together, and the region where a profit is made is shaded. The optimum profit occurs at the endpoints (or vertices) of the region. EXAMPLE A compan makes both roller skates ( X ) and ice skates ( Y ). Roller skates make a $5 profit, while ice skates make a profit of $. Each pair of roller skates spends hours on machine A (available hours per da) and hours on machine B (available 8 hours per da). Each pair of ice skates spends hours on machine A and hour on machine B. How man skates of each tpe should be made each da to give the greatest profit while making the most efficient use of the machines? SOLUTION Profit P = $5 X+ $Y Machine A: X + Y # Machine B: X + Y # 8 Sketch the regions and find the point of intersection of the lines. The shaded area shows all possible was of making a profit. Optimum profit occurs at one of the endpoints of the regions. (0, ): P = $5] 0 g + $] g = $8 (, 0): P = $5] g + $] 0 g = $00 (, ): P = $5] g + $] g = $7 ^, h gives the greatest profit, so pairs of roller skates and pairs of ice skates each da gives optimum profit.

84 Chapter 5 Functions and Graphs 87 Test Yourself 5. If f] g = - -, find (a) f ]-g (b) f ] ag (c) when f ] g = 0. Sketch each graph (a) = - - (b) f ] g = (c) + = (d) = - (e) =- - (f) = (g) = 0 (h) = +. Describe each region (a) (b). Find the domain and range of each graph in question. if $. If f] g = * - if find f] 5g- f ] 0g+ f ] g if 5. Given f] g = * if # # find (a) f ] g (b) f ]-g (c) f ] g (d) f ] 5g (e) f ] 0g - if (c) 6. Shade the region $ Shade the region where and $ Shade the region given b + $. 9. Shade the region given b # 0and $ -.. (a) Write down the domain and range of the curve =. - (b) Sketch the graph of = Shade the region + and + #.