5.3 Graphing Cubic Functions

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1 Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1 Graphing and Analzing f () = 3 You know that a quadratic function has the standard form ƒ () = a + b + c where a, b, and c are real numbers and a. Similarl, a cubic function has the standard form ƒ () = a 3 + b + c + d where a, b, c and d are all real numbers and a. You can use the basic cubic function, ƒ () = 3, as the parent function for a famil of cubic functions related through transformations of the graph of ƒ () = 3. A Complete the table, graph the ordered pairs, and then draw a smooth curve through the plotted points to obtain the graph of ƒ () = 3. = B Use the graph to analze the function and complete the table Domain Range Attributes of f () = 3 R -6-8 Houghton Mifflin Harcourt Publishing Compan End behavior As +, f (). As -, f (). Zeros of the function = Where the function has positive values > Where the function has negative values Where the function is increasing Where the function is decreasing The function never decreases. Is the function even (f (-) = f () ), odd (f (-) = -f () ), or neither?, because () 3 =. Module 5 79 Lesson 3

2 Reflect 1. How would ou characterize the rate of change of the function on the intervals -1, and, 1 compared with the rate of change on the intervals -, -1 and 1,? Eplain.. A graph is said to be smmetric about the origin (and the origin is called the graph s point of smmetr) if for ever point (, ) on the graph, the point (-, -) is also on the graph. Is the graph of ƒ () = 3 smmetric about the origin? Eplain. 3. The graph of g () = (-) 3 is a reflection of the graph of ƒ () = 3 across the -ais, while the graph of h () = - 3 is a reflection of the graph of ƒ () = 3 across the -ais. If ou graph g() and h() on a graphing calculator, what do ou notice? Eplain wh this happens. Eplain 1 Graphing Combined Transformations of f () = 3 When graphing transformations of ƒ () = 3, it helps to consider the effect of the transformations on the three reference points on the graph of ƒ () : (-1, -1), (, ), and (1,1). The table lists the three points and the corresponding points on the graph of g () = a ( 1 b ( - h) ) 3 + k. Notice that the point (, ), which is the point of smmetr for the graph of ƒ (), is affected onl b the parameters h and k. The other two reference points are affected b all four parameters. f () = 3 g () = a ( 1_ b ( - h) ) 3 + k b + h -a + k h k 1 1 b + h a + k Houghton Mifflin Harcourt Publishing Compan Module 5 8 Lesson 3

3 Eample 1 Identif the transformations of the graph of ƒ () = 3 that produce the graph of the given function g (). Then graph g () on the same coordinate plane as the graph of ƒ () b appling the transformations to the reference points (-1, -1), (, ), and (1, 1). A g () = ( - 1) 3-1 The transformations of the graph of ƒ () that produce the graph of g () are: a vertical stretch b a factor of a translation of 1 unit to the right and 1 unit down Note that the translation of 1 unit to the right affects onl the -coordinates of points on the graph of ƒ (), while the vertical stretch b a factor of and the translation of 1 unit down affect onl the -coordinates. f () = 3 g () = ( - 1) = (-1) - 1 = = 1 () - 1 = = (1) - 1 = 1 B g () = ( ( + 3) ) 3 + The transformations of the graph of ƒ () that produce the graph of g () are: 6 Houghton Mifflin Harcourt Publishing Compan a horizontal compression b a factor of 1 a translation of 3 units to the left and units up Note that the horizontal compression b a factor of 1 and the translation of 3 units to the left affect onl the -coordinates of points on the graph of ƒ (), while the translation of units up affects onl the -coordinates. f () = 3 g () = ( ( + 3) ) (-1) + = -1 + = () + = + = 1 1 (1) + = 1 + = Module 5 81 Lesson 3

4 Your Turn Identif the transformations of the graph of ƒ () = 3 that produce the graph of the given function g (). Then graph g () on the same coordinate plane as the graph of ƒ () b appling the transformations to the reference points (-1, -1), (, ), and (1, 1).. g () = -_ 1 3 ( - 3) Eplain Writing Equations for Combined Transformations of f () = 3 Given the graph of the transformed function g () = a ( 1 b ( - h) ) 3 + k, ou can determine the values of the parameters b using the same reference points that ou used to graph g () in the previous eample. Eample A general equation for a cubic function g () is given along with the function s graph. Write a specific equation b identifing the values of the parameters from the reference points shown on the graph. A g () = a ( - h) 3 + k Identif the values of h and k from the point of smmetr. (3, ) (h, k) = (, 1), so h = and k = 1. Identif the value of a from either of the other two reference points. The rightmost reference point has general coordinates (h + 1, a + k). Substituting for h and 1 for k and setting the general coordinates equal to the actual coordinates gives this result: (h + 1, a + k) = (3, a + 1) = (3, ), so a = 3. Write the function using the values of the parameters: g () = 3 ( - ) (, 1) 6 (1, -) Houghton Mifflin Harcourt Publishing Compan Module 5 8 Lesson 3

5 B g () = ( 1 b - h ) 3 + k Identif the values of h and k from the point of smmetr. (h, k) = (-, ), so h = - and k =. Identif the value of b from either of the other two reference points. The rightmost reference point has general coordinates (b + h, 1 + k). Substituting - for h and for k and setting the general coordinates equal to the actual coordinates gives this result: ( b + h, 1 + ) = (b -, ) = (-3.5, ), so b =. a = 1/(1/) a = (-3.5, ) (-.5, ) (-, 1) = ( + ) Write the function using the values of the parameters, and then simplif. ( g () 1 = ( - )) 3 + or g () = ( ( + ) ) 3 + Your Turn A general equation for a cubic function g () is given along with the function s graph. Write a specific equation b identifing the values of the parameters from the reference points shown on the graph. 5. g () = a ( - h) 3 + k 6. g () = ( 1 _ b ( - h) ) 3 + k Houghton Mifflin Harcourt Publishing Compan (-1, 1) (-, -) (-3, -5) (-3, -) - (5, ) (1, -1) Module 5 83 Lesson 3

6 Evaluate: Homework and Practice 1. Graph the parent cubic function ƒ () = 3 and use the graph to answer each question. a. State the function s domain and range. Domain: b. Identif the function s end behavior. As -> +, f() - > As -> -, f() - > Range: c. Identif the graph s - and -intercepts. d. Identif the intervals where the function has positive values and where it has negative values e. Identif the intervals where the function is increasing and where it is decreasing. -8 f. Tell whether the function is even, odd, or neither. Eplain. g. Describe the graph s smmetr. Houghton Mifflin Harcourt Publishing Compan Describe how the graph of g() is related to the graph of ƒ () = 3.. g () = ( - ) 3 3. g () = g () = g () = (3) 3 Module 5 87 Lesson 3

7 Int Math 3 pp.87-88!

8 6. g () = ( + 1) 3 7. g () = 1 _ 3 8. g () = g () = ( - _ 3 ) 3 Identif the transformations of the graph of ƒ () = 3 that produce the graph of the given function g(). Then graph g() on the same coordinate plane as the graph of ƒ () b appling the transformations to the reference points (-1, -1), (, ), and (1, 1). 1. g () = ( 1 _ 3 ) g () = 1 _ g () = ( - ) g () = ( + 1) Houghton Mifflin Harcourt Publishing Compan Module 5 88 Lesson 3

9 Int Math 3 pp.87-88

10 A general equation for a cubic function g () is given along with the function s graph. Write a specific equation b identifing the values of the parameters from the reference points shown on the graph. 1. g () = ( 1 _ b ( - h) ) 3 + k (-, -3) - - (-5, -) -6 (1, -) 15. g () = a ( - h) 3 + k 6 (, 7) (1, ) (, 1) g () = ( 1 _ b ( - h) ) 3 + k - (, -1) - - (1.5, ) 6 (.5, -) 17. g () = a ( - h) 3 + k (-, 1.5) (-1, ) (,.5) Houghton Mifflin Harcourt Publishing Compan Module 5 89 Lesson 3

11 . Multiple Response Select the transformations of the graph of the parent cubic function that result in the graph of g () = (3 ( - ) ) A. Horizontal stretch b a factor of 3 E. Translation 1 unit up B. Horizontal compression b a factor of 1 _ 3 F. Translation 1 unit down C. Vertical stretch b a factor of 3 G. Translation units left 1_ D. Vertical compression b a factor of 3 H. Translation units right H.O.T. Focus on Higher Order Thinking 1. Justif Reasoning Eplain how horizontall stretching (or compressing) the graph of ƒ () = 3 b a factor of b can be equivalent to verticall compressing (or stretching) the graph of ƒ () = 3 b a factor of a. Houghton Mifflin Harcourt Publishing Compan. Critique Reasoning A student reasoned that g () = ( - h) 3 can be rewritten as g () = 3 - h 3, so a horizontal translation of h units is equivalent to a vertical translation of - h 3 units. Is the student correct? Eplain. Module 5 91 Lesson 3