1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr, where r is the radius. The volume, V, of helium in a spherical balloon can be modelled b the function V(r) _ 3 πr 3, where r is again the radius. The functions that represent each situation are called power functions. A power function is the simplest tpe of polnomial function and has the form f() a n, where is a variable, a is a real number, and n is a whole number. CONNECTIONS Polnomials are the building blocks of algebra. Polnomial functions can be used to create a variet of other tpes of functions and are important in man areas of mathematics, including calculus and numerical analsis. Outside mathematics, the basic equations in economics and man phsical sciences are polnomial equations. A polnomial epression is an epression of the form a n n a n 1 n 1 a n n... a 3 3 a a 1 a, where n is a whole number is a variable the coefficients a, a 1,..., a n are real numbers the degree of the function is n, the eponent of the greatest power of a n, the coefficient of the greatest power of, is the leading coefficient a, the term without a variable, is the constant term A polnomial function has the form f() a n n a n 1 n 1 a n n... a 3 3 a a 1 a Polnomial functions are tpicall written in descending order of powers of. The eponents in a polnomial do not need to decrease consecutivel; that is, some terms ma have zero as a coefficient. For eample, f() 3 1 is still a polnomial function even though there is no -term. A constant function, of the form f() a, is also a tpe of polnomial function (where n ), as ou can write the constant term a in the form a. MHR Advanced Functions Chapter 1
Investigate What are the ke features of the graphs of power functions? 1. Match each graph with the corresponding function. Justif our choices. Use a graphing calculator if necessar. a) b) c) 3 d) e) 5 f) 6 i) ii) 96 6 3 3 6 96 16 18 96 6 3 3 Tools graphing calculator CONNECTIONS Some power functions have special names that are associated with their degree. Power Function Degree Name a constant a 1 linear a quadratic a 3 3 cubic a quartic a 5 5 quintic iii) 16 iv) 18 96 6 3 3 v) vi) 8 6 1.1 Power Functions MHR 5
CONNECTIONS Recall that a relation is a function if for ever -value there is onl one -value. The graph of a relation represents a function if it passes the vertical line test, that is, if a vertical line drawn anwhere along the graph intersects that graph at no more than one point. The end behaviour of the graph of a function is the behaviour of the -values as increases (that is, as approaches positive infinit, written as ) and as decreases (that is, as approaches negative infinit, written as ). CONNECTIONS A graph has line smmetr if there is a line a that divides the graph into two parts such that each part is a reflection of the other in the line a. a A graph has point smmetr about a point (a, b) if each part of the graph on one side of (a, b) can be rotated 18 to coincide with part of the graph on the other side of (a, b). a, b. a) Reflect Decide whether each graph in step 1 represent a linear, a quadratic, a cubic, a quartic, or a quintic function. Justif our answer. b) Reflect Eplain wh each graph in step 1 represents a function. 3. a) State these ke features for each graph: i) the domain ii) the range iii) the intercepts b) Describe the end behaviour of each graph as i) ii). a) Which graphs have both ends etending upward in quadrants 1 and (that is, start high and end high)? b) Decide whether each graph has line smmetr or point smmetr. Eplain. c) Reflect Describe how the graphs are similar. How are the equations similar? 5. a) Which graphs have one end etending downward in quadrant 3 (start low) and the other end etending upward in quadrant 1 (end high)? b) Decide whether each graph has line smmetr or point smmetr. Eplain. c) Reflect Describe how the graphs are similar. How are the equations similar? 6. Reflect Summarize our findings for each group of power functions in a table like this one. Ke Features of the Graph n, n is odd n, n is even Domain Range Smmetr End Behaviour 7. a) Graph the function n for n,, and 6. b) Describe the similarities and differences between the graphs. c) Reflect Predict what will happen to the graph of n for larger even values of n. d) Check our prediction in part c) b graphing two other functions of this form. 8. a) Graph the function n for n 1, 3, and 5. b) Describe the similarities and differences between the graphs. c) Reflect Predict what will happen to the graph of n for larger odd values of n. d) Check our prediction in part c) b graphing two more functions of this form. 6 MHR Advanced Functions Chapter 1
Eample 1 Recognize Polnomial Functions Determine which functions are polnomials. Justif our answer. State the degree and the leading coefficient of each polnomial function. a) g() sin b) f() c) 3 5 6 8 d) g() 3 Solution a) g() sin This a trigonometric function, not a polnomial function. b) f() This is a polnomial function of degree. The leading coefficient is. c) 3 5 6 8 This is a polnomial function of degree 3. The leading coefficient is 1. d) g() 3 This is not a polnomial function but an eponential function, since the base is a number and the eponent is a variable. Interval Notation In this course, ou will often describe the features of the graphs of a variet of tpes of functions in relation to real-number values. Sets of real numbers ma be described in a variet of was: as an inequalit, 3 5 in interval (or bracket) notation ( 3, 5] 3 1 1 3 5 6 graphicall, on a number line Intervals that are infinite are epressed using the smbol (infinit) or (negative infinit). Square brackets indicate that the end value is included in the interval, and round brackets indicate that the end value is not included. A round bracket is used at infinit since the smbol means without bound. 1.1 Power Functions MHR 7
Below is a summar of all possible intervals for real numbers a and b, where a b. Bracket Interval Inequalit Number Line In Words The set of all real numbers such that (a, b) a b a b is greater than a and less than b (a, b] a b is greater than a and less than a b or equal to b [a, b) a b is greater than or equal to a and a b less than b [a, b] a b is greater than or equal to a and a b less than or equal to b [a, ) a a is greater than or equal to a (, a] a a is less than or equal to a (a, ) a a is greater than a (, a) a a is less than a (, ) is an element of the real numbers Eample Connect the Equations and Features of the Graphs of Power Functions For each function i) state the domain and range ii) describe the end behaviour iii) identif an smmetr a) b) 8.5 6 c) 1 6 3 6 1 8 MHR Advanced Functions Chapter 1
Solution a) i) domain { } or (, ); range { } or (, ) ii) The graph etends from quadrant to quadrant. Thus, as,, and as,. iii) The graph has point smmetr about the origin (, ). b).5 i) domain { } or (, ); range {, } or [, ) ii) The graph etends from quadrant to quadrant 1. Thus, as,, and as,. iii) The graph has line smmetr in the -ais. c) 3 i) domain { } or (, ); range { } or (, ) ii) The graph etends from quadrant 3 to quadrant 1. Thus, as,, and as,. iii) The graph has point smmetr about the origin. Eample 3 Describe the End Behaviour of Power Functions Write each function in the appropriate row of the second column of the table. Give reasons for our choices. 5 6 3 7 _ 5 9 5 1.5 8 End Behaviour Function Reasons Etends from quadrant 3 to quadrant 1 Etends from quadrant to quadrant Etends from quadrant to quadrant 1 Etends from quadrant 3 to quadrant Solution End Behaviour Function Reasons Etends from quadrant 3 to quadrant 1, 7 odd eponent, positive coefficient Etends from quadrant to quadrant _ 5 9 5, odd eponent, negative coefficient Etends from quadrant to quadrant 1 5 6, 1 even eponent, positive coefficient Etends from quadrant 3 to quadrant 3,.5 8 even eponent, negative coefficient 1.1 Power Functions MHR 9
Eample Connecting Power Functions and Volume Helium is pumped into a large spherical balloon designed to advertise a new product. The volume, V, in cubic metres, of helium in the balloon is given b the function V(r) _ 3 πr 3, where r is the radius of the balloon, in metres, and r [, 5]. Reasoning and Proving Representing Selecting Tools Problem Solving Connecting Reflecting Communicating a) Graph V(r). b) State the domain and range in this situation. c) Describe the similarities and differences between the graph of V(r) and the graph of 3. Solution a) Make a table of values, plot the points, and connect them using a smooth curve. V(r) _ 3 πr 3 r (m) (m 3 ) _ 3 π()3 1 _ 3 π(1)3. _ 3 π()3 33.5 3 _ 3 π(3)3 113.1 _ 3 π()3 68.1 5 _ 3 π(5)3 53.6 Volume (m 3 ) V 6 1 3 5 Radius (m) r b) The domain is r [, 5]. The range is approimatel V [, 53.6]. c) The graph of 3 is shown. Similarities: The functions V(r) _ 3 πr3 and 3 are both cubic, with positive leading coefficients. Both graphs pass 3 through the origin (, ) and have one 6 6 end that etends upward in quadrant 1. Differences: the graph of V(r) has a restricted domain. Since the two functions are both cubic power functions that have different leading coefficients, all points on each graph, other than (, ), are different. 1 MHR Advanced Functions Chapter 1
<< >> KEY CONCEPTS A polnomial epression has the form a n n a n 1 n 1 a n n... a 3 3 a a 1 a where n is a whole number is a variable the coefficients a, a 1,..., a n are real numbers the degree of the function is n, the eponent of the greatest power of a n, the coefficient of the greatest power of, is the leading coefficient a, the term without a variable, is the constant term A polnomial function has the form f() a n n a n 1 n 1 a n n... a a 1 a A power function is a polnomial of the form a n, where n is a whole number. Power functions have similar characteristics depending on whether their degree is even or odd. Even-degree power functions have line smmetr in the -ais,. Odd-degree power functions have point smmetr about the origin, (, ). Communicate Your Understanding C1 C C3 C Eplain wh the function 3 is a polnomial function. How can ou use a graph to tell whether the leading coefficient of a power function is positive or negative? How can ou use a graph to tell whether the degree of a power function is even or odd? State a possible equation for a power function whose graph etends a) from quadrant 3 to quadrant 1 b) from quadrant to quadrant 1 c) from quadrant to quadrant d) from quadrant 3 to quadrant A Practise For help with questions 1 and, refer to Eample 1. 1. Identif whether each is a polnomial function. Justif our answer. a) p() cos b) h() 7 c) f() d) 3 5 3 1 e) k() 8 f) 3. State the degree and the leading coefficient of each polnomial. a) 5 3 3 b) c) 8 d) _ 3 3 e) 5 f) 3 1.1 Power Functions MHR 11
For help with question 3, refer to Eample. d) 3. Consider each graph. i) Does it represent a power function of even degree? odd degree? Eplain. ii) State the sign of the leading coefficient. Justif our answer. iii) State the domain and range. iv) Identif an smmetr. e) v) Describe the end behaviour. a) For help with question, refer to Eample 3. b). Cop and complete the table for the following functions. 3 _ 3 7 5 5 6.1 11 9 1 End Behaviour Function Reasons c) Etends from quadrant 3 to quadrant 1 Etends from quadrant to quadrant Etends from quadrant to quadrant 1 Etends from quadrant 3 to quadrant B Connect and Appl For help with questions 5 and 6, refer to Eample. 5. As a tropical storm intensifies and reaches hurricane status, it takes on a circular shape that epands outward from the ee of the storm. The area, A, in square kilometres, spanned b a storm with radius, r, in kilometres, can be modelled b the function A(r) πr. a) Graph A(r) for r [, 1]. b) State the domain and range. c) Describe the similarities and differences between the graph of A(r) and the graph of. 6. The circumference, C, in kilometres, of the tropical storm in question 5 can be modelled b the function C(r) πr. a) Graph C(r) for r [, 1]. b) State the domain and range. c) Describe the similarities and differences between the graph of C(r) and the graph of. 1 MHR Advanced Functions Chapter 1
7. Determine whether each graph represents a power function, an eponential function, a periodic function, or none of these. Justif our choice. g) CONNECTIONS You worked with periodic functions when ou studied trigonometric functions in grade 11. Periodic functions repeat at regular intervals. a) b) c) d) e) f) 8. Use Technolog a) Graph f() 3, g() 3, and Problem Solving h() 3 on the Connecting Communicating same set of aes. b) Compare and describe the ke features of the graphs of these functions. 9. Use Technolog a) Graph f(), g(), and h() on the same set of aes. b) Compare and describe the ke features of the graphs of these functions. 1. Describe the similarities and differences between the line and power functions with odd degree greater than one. Use graphs to support our answer. 11. Describe the similarities and differences between the parabola and power functions with even degree greater than two. Use graphs to support our answer. 1. a) Graph the functions Reasoning and Proving 3, 3 Representing Selecting Tools, and 3 on Problem Solving the same set of aes. Connecting Reflecting Communicating Compare the graphs. Describe how the graphs are related. b) Repeat part a) for the functions,, and. c) Make a conjecture about the relationship between the graphs of n and n c, where c and n is a whole number. d) Test the accurac of our conjecture for different values of n and c. Representing Reasoning and Proving Selecting Tools Reflecting 1.1 Power Functions MHR 13
13. Chapter Problem Part of a computer graphic designer s job ma be to create and manipulate electronic images found in video games. Power functions define curves that are useful in the design of characters, objects, and background scener. Domain restrictions allow two or more curves to be combined to create a particular form. For eample, a character s ee could be created using parabolas with restricted domains. Describe the tpe(s) of power function(s) that could be used to design two of the following in a new video game. Provide equations and sketches of our functions. Include the domain and range of the functions ou use. the path of a river that etends from the southwest to the northeast part of a large forest the cross-section of a valle that lies between two mountain ranges a deep canon where the river flows characters facial epressions a lightning bolt horseshoe tracks in the sand C Etend and Challenge 1. Use Technolog a) Graph each pair of functions. What do ou notice? Provide an algebraic eplanation for what ou observe. i) ( ) and ii) ( ) and iii) ( ) 6 and 6 b) Repeat part a) for each of the following pairs of functions. i) ( ) 3 and 3 ii) ( ) 5 and 5 iii) ( ) 7 and 7 c) Describe what ou have learned about functions of the form ( ) n, where n is a non-negative integer. Support our answer with eamples. 15. a) Make a conjecture about the relationship between the graphs of n and a n for a. b) Test our conjecture using different values of n and a. 16. a) Describe the relationship between the graph of and the graph of ( 3) 1. b) Predict the relationship between the graph of and the graph of ( 3) 1. c) Verif the accurac of our prediction b sketching the graphs in part b). 17. a) Use the results of question 16 to predict a relationship between the graph of 3 and the graph of a( h) 3 k. b) Verif the accurac of our prediction in part a) b sketching two functions of the form a( h) 3 k. 18. Math Contest Determine the number of digits in the epansion of ( 1 )(5 15 ) without using a calculator or computer. 19. Math Contest Find the coordinates of the two points that trisect the line segment with endpoints A(, 3) and B(8, ). 1 MHR Advanced Functions Chapter 1