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 68 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) 31 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.8788!
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.8788
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
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