Graphing Cubic Functions

Transcription

1 Locker LESSON. Graphing Cubic Functions Name Class Date. Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) + k and f () = ( related to the graph of f () =? b ( - h) ) + k Teas Math Standards The student is epected to: A..A Graph the functions f ( ) =, f ( ) =, f ( ) =, f () =, f ( ) = b, f ( ) =, and f ( ) = log b () where b is,, and e, and, when applicable, analze the ke attributes such as domain, range, intercepts, smmetries, asmptotic behavior, and maimum and minimum given an interval. Also A..A Mathematical Processes A..D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Language Objective.C.,.B.,.C.,.H. Eplain to a partner how to predict transformations of a basic cubic function. ENGAGE Essential Question: How are the graphs of f () = a ( - h) + k and f () = ( b ( - h) ) + k related to the graph of f () =? Possible answer: Both graphs involve transformations of the graph of f() =. The first involves verticall stretching or compressing the graph of f() =, reflecting it across the -ais if a <, and translating it horizontall and verticall. The second involves horizontall stretching or compressing the graph of f() =, reflecting it across the -ais if b <, and translating it horizontall and verticall. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the volume of a spherical aquarium can be described b a cubic function. Then preview the Lesson Performance Task. 5 Lesson. Houghton Mifflin Harcourt Publishing Compan A..A...analze ke attributes such as domain, range, intercepts, smmetries, asmptotic behavior, and maimum and minimum given an interval. Also A..A. Eplore Graphing and Analzing f() = Resource Locker 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 + b + c + d where a, b, c and d are all real numbers and a. You can use the basic cubic function, ƒ () =, as the parent function for a famil of cubic functions related through transformations of the graph of ƒ () =. Complete the table, graph the ordered pairs, and then draw a smooth curve through the plotted points to obtain the graph of ƒ () =. Use the graph to analze the function and complete the table. Domain Range Attributes of f () = 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 R R = The function increases throughout its domain. The function never decreases. Is the function even (f(-) = f ()), odd Odd (f(-) = -f()), or neither?, because () = -. Module 5 Lesson DO NOT EDIT--Changes must be made through "File info" CorrectionKe=TX-B Name Class Date. Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) + k and f () = ( b ( - h) ) + k related to the graph of f () =? A..A...analze ke attributes such as domain, range, intercepts, smmetries, asmptotic behavior, and maimum and minimum given an interval. Also A..A. Eplore Graphing and Analzing f() = Houghton Mifflin Harcourt Publishing Compan Resource 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 + b + c + d where a, b, c and d are all real numbers and a. You can use the basic cubic function, ƒ () =, as the parent function for a famil of cubic functions related through transformations of the graph of ƒ () =. Complete the table, graph the ordered pairs, and then draw a smooth curve through the plotted points to obtain the graph of ƒ () =. Use the graph to analze the function and complete the table. Attributes of f () = = -8-8 Domain R Range 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 () =. R - - < The function increases throughout its domain. Odd + - Module 5 Lesson A_MTXESE59_UML.indd :5 AM - HARDCOVER PAGES 5 Turn to these pages to find this lesson in the hardcover student edition.

2 Reflect. How would ou characterize the rate of change of the function on the intervals -, and, compared with the rate of change on the intervals -, - and,? Eplain. The rate of change on all four intervals is positive because the function is increasing. However, the rate of change on the intervals -, and, is much less than the rate of change on the intervals -, - and,, because the graph rises more slowl on the intervals -, and, than it does on the intervals -, - and,.. 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 () = smmetric about the origin? Eplain. Yes, because for ever point (, ) = (, ) on the graph of the function, the point (-, (-) ) = (-, - ) = (-, -) is also on the graph. Eplore Predicting Transformations of the Graph of f () = You can appl familiar transformations to cubic functions. Check our predictions using a graphing calculator. A Predict the effect of the parameter h on the graph of g () = ( - h) for each function. a. The graph of g () = ( - ) translation is a of the graph of ƒ () [right/up/left/down] units. b. The graph of g () = ( + ) translation is a of the graph of ƒ () [right/up/left/down] units. B Predict the effect of the parameter k on the graph of g () = + k for each function. translation a. The graph of g () = + is a of the graph of ƒ () [right/up/left/down] units. translation b. The graph of g () = - is a of the graph of ƒ () [right/up/left/down] units. C Predict the effect of the parameter a on the graph of g () = a for each function. a. The graph of g () = is a vertical stretch of the graph of ƒ () b a factor of. b. The graph of g () = - vertical compression is a of the graph of ƒ () b a factor of reflection as well as a across -ais the. Module Lesson PROFESSIONAL DEVELOPMENT Math Background A cubic function is a polnomial function of degree. The standard form of a cubic function is f () = a + b + c + d where a, b, c, and d are real numbers and a. Each function of the famil of cubic functions is increasing or decreasing over its entire domain, depending on the sign of the leading coefficient of. There also ma be turning points and relative maimum and relative minimum points, depending on the transformations of the graph of f () =. The domain and range of cubic functions are the set of all real numbers. The inverse of a cubic function ma or ma not be a function. If the graph has turning points, its inverse is not a function. Houghton Mifflin Harcourt Publishing Compan EXPLORE Graphing and Analzing f () = INTEGRATE TECHNOLOGY Students can use a graphing calculator to graph and generate table values for the function =. QUESTIONING STRATEGIES How do ou verif that the graph of f() = has smmetr about the origin? Show that if (, ) is a point on the graph, then (-, -) is also a point on the graph. How is the graph of f() = similar to the graph of f() =? The domain and range for both functions are (-, ) ; the end behavior for both functions is f () - as -, and f () as +. EXPLORE Predicting Transformations of the Graph of f () = AVOID COMMON ERRORS Students often assume that the graph of f( - h) is translated to the left due to the subtraction. For eample, if the function is f() = ( - ), students ma want to translate f() = two units to the left. Point out that the value of h in f( - h) is positive, so the transformation is in the positive direction, to the right. Have students graph the two functions to verif this. CONNECT VOCABULARY Relate the term parent function to parents and children. The children come from the parents; likewise, cubic functions can all be traced back to a parent function. Graphing Cubic Functions

3 QUESTIONING STRATEGIES How is the graph of g() = ( - h ) + k related to the parent graph f() =? The graph of g () = ( - h) + k is the graph of f () = translated h units horizontall and k units verticall. If a >, how is the graph of g() = a related to the graph of f() =? If a >, the graph of g() is the graph of f () = stretched a units verticall. If < a <, the graph of g() is the graph of f() = compressed units verticall. a EXPLAIN Graphing Combined Transformations of f () = INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Point out that to graph the function f () = a ( - h) + k, ou identif the point of smmetr (h, k) and use the value of a to draw the graph through two additional points (- + h, a + k) and ( + h, a + k). So, to graph f () = ( - ) +, first identif the point of smmetr, (, ). Then identif points (- +, - + ) = (, -), and ( +, + ) = (, ) as additional reference points. A smooth curve through these three points is a good beginning for the graph. Houghton Mifflin Harcourt Publishing Compan D Predict the effect of the parameter b on the graph of g () = ( b ) for the following values of b. a. The graph of g () = ( ) is a horizontal stretch of the graph of ƒ () b a factor of. b. The graph of g () = () is a horizontal compression of the graph of ƒ () b a factor of. c. The graph of g () = (- ) is a horizontal compression of the graph of ƒ () b a factor of as well as a reflection across the -ais. d. The graph of g () = (-) is a horizontal stretch of the graph of ƒ () b a Reflect factor of as well as a reflection across the -ais.. The graph of g () = (-) is a reflection of the graph of ƒ () = across the -ais, while the graph of h () = - is a reflection of the graph of ƒ () = across the -ais. If ou graph g() and h() on a graphing calculator, what do ou notice? Eplain wh this happens. The graphs of g () and h () are the same because f () is an odd function (that is, f (-) = -f () ). Given that g () = f (-) and h () = -f (), ou have g () = h (). Eplain Graphing Combined Transformations of f () = When graphing transformations of ƒ () =, it helps to consider the effect of the transformations on the three reference points on the graph of ƒ () : (-, -), (, ), and (,). The table lists the three points and the corresponding points on the graph of g () = a ( b ( - h) ) + 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 () = g () = a ( b ( - h) ) + k - - -b + h -a + k h k b + h a + k Module 7 Lesson COLLABORATIVE LEARNING Whole Class Activit Have groups of students create posters to describe the transformations to the graph of f () = when graphing f () = a ( - h) + k. Have them write an eample function in the top cell of a graphic organizer. Then, in each cell below the eample function, have them list the variable and describe the nature of the transformation. 7 Lesson.

4 Eample Identif the transformations of the graph of ƒ () = 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 (-, -), (, ), and (, ). g () = ( - ) - The transformations of the graph of ƒ () that produce the graph of g () are: a vertical stretch b a factor of a translation of unit to the right and unit down - - QUESTIONING STRATEGIES What is the point of smmetr for a graph of the form f() = ( ( b - h) ) + k? (h, k) How does a negative value for b affect the graph of f() = ( ( b - h) ) + k? The graph is reflected across the -ais. Note that the translation of 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 unit down affect onl the -coordinates. - f () = g () = ( - ) = (-) - = - + = () - = - + = () - = g () = ( ( + ) ) + The transformations of the graph of ƒ () that produce the graph of g () are: a horizontal compression b a factor of a translation of units to the left and units up Note that the horizontal compression b a factor of and the translation of 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 () = g () = ( ( + ) ) (-) + - = - + = () + - = - + = - () + - = + = Houghton Mifflin Harcourt Publishing Compan Module 8 Lesson DIFFERENTIATE INSTRUCTION Critical Thinking Have students form pairs. Give each pair of students a cubic function in general form and a graphing calculator. Have them graph the parent function f () = and then discuss how their given cubic function will be transformed based the graph of the parent function. Include choices such as reflection across the -ais, reflection across the -ais, horizontal and vertical shifts, and horizontal and vertical stretches. Have them graph their given functions to verif their predictions. Graphing Cubic Functions 8

5 EXPLAIN Writing Equations for Combined Transformations of f () = AVOID COMMON ERRORS For students having difficult identifing the parameters in the graph of g () = a ( b ( - h) ) + k, have them start with identifing the images of the reference points (-, - ), (, ), (, ) from the graph of f() =. Then have them solve for the parameters a, b, h, and k using the image points and (-b + h, -a + k), (h, k), or (b + h, a + k), respectivel. Your Turn Identif the transformations of the graph of ƒ () = 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 (-, -), (, ), and (, ).. g () = - ( - ) The transformations of the graph of f () that produce the graph of g () are: a vertical compression b a factor of a reflection across the -ais a translation of units to the right - Note that the translation of units to the right affects onl the -coordinates of points on the graph of f (), while the vertical compression b a factor of and the reflection across the -ais affect onl the -coordinates. Points: (, ), (, ), and (, - ) Eplain Writing Equations for Combined Transformations of f () = Given the graph of the transformed function g () = a ( b ( - h) ) + 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. g () = a ( - h) + k Identif the values of h and k from the point of smmetr. (, ) Houghton Mifflin Harcourt Publishing Compan (h, k) = (, ), so h = and k =. Identif the value of a from either of the other two reference points. The rightmost reference point has general coordinates (h +, a + k). Substituting for h and for k and setting the general coordinates equal to the actual coordinates gives this result: (h +, a + k) = (, a + ) = (, ), so a =. Write the function using the values of the parameters: g () = ( - ) (, ) (, -) Module 9 Lesson LANGUAGE SUPPORT Connect Vocabular Have students work in pairs. Instruct one student to eplain the effect of an transformation of the parent function f() =. While the first student eplains, the second student writes down the steps in the eplanation. The switch roles and repeat the procedure for another change, such as the effect of b on the graph of g () = b. The goal is to support students as the learn to articulate their understanding at first, using their own words, then moving towards more mathematicall precise eplanations. 9 Lesson.

6 B g () = ( b - h ) + 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, + k). Substituting - for h and for k and setting the general coordinates equal to the actual coordinates gives this result: ( b + h, + ) = (b -, ) = (-.5, ), so b =.5. Write the function using the values of the parameters, and then simplif. g () = ( or ( - ).5 ) g () = ( ( + ) ) (-.5, ) (-.5, ) (-, ) QUESTIONING STRATEGIES Wh does the general form f () = a ( - h) + k or f () = ( ( b - h) ) + k need to be specified before finding the equation of the transformed function? The general form must be specified so that a unique equation can be found. 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) + k. g () = ( b ( - h) ) + k (-, ) (-, -) (-, -5) (h, k) = (-, -), so h = - and k = -. (h +, a + k) = (- +, a - ) = (-, ), so a =. g () = ( + ) (-, -) - (5, ) (, -) (h, k) = (, -), so h = and k = -. (b + h, + k) = (b +, - ) = (5, ), so b =. g () = ( ( - ) ) - Houghton Mifflin Harcourt Publishing Compan Module Lesson Graphing Cubic Functions

7 EXPLAIN Modeling with a Transformation of f ( ) = QUESTIONING STRATEGIES How is the densit formula for a sphere of radius r, m (r) = DV (r) with mass m, densit D, and volume V related to the volume of a sphere, V(r) = π r, with radius r? The mass of the sphere is the product of the densit and the volume, so m (r) = πdr. What kind of transformation does the equation m (r) = πdr show? a vertical stretch of the function m (r) = r b a factor of πd Eplain Modeling with a Transformation of f () = You ma be able to model a real-world situation that involves volume with a cubic function. Sometimes mass ma also be involved in the problem. Mass and volume are related through densit, which is defined as an object s mass per unit volume. If an object has mass m and volume V, then its densit d is d = m. You can rewrite the formula as m = V dv to epress mass in terms of densit and volume. Eample Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate. Estimate the length of an edge of a child s alphabet block (a cube) that has a mass of g and is made from oak with a densit of.7 g/c m. Let l represent the length (in centimeters) of an edge of the block. Since the block is a cube, the volume V (in cubic centimeters) is V (l) = l. The mass m (in grams) of the block is m (l) =.7 V (l) =.7 l. Make a table of values for this function. Length (cm) Mass (g) Houghton Mifflin Harcourt Publishing Compan Image Credits: MidoSemsem/Shutterstock Draw the graph of the mass function, recognizing that the graph is a vertical compression of the graph of the parent cubic function b a factor of.7. Then draw the horizontal line m = and estimate the value of l where the graphs intersect. Mass (g) m l Length (cm) The graphs intersect where l., so the edge length of the child s block is about. cm. Module Lesson Lesson.

8 B Estimate the radius of a steel ball bearing with a mass of 75 grams and a densit of 7.8 g/c m. Let r represent the radius (in centimeters) of the ball bearing. The volume V (in cubic centimeters) of the ball bearing is _ V (r) = r. The mass m (in grams) of the ball π bearing is m (r) = 7.8 V (r) =.7 r. Radius (cm).5.5 Mass (g) Mass (g) stretch Draw the graph of the mass function, recognizing that the graph is a vertical of the graph of the parent cubic function b a factor of 7.8. Then draw the m.5.5 Radius (cm) r horizontal line m = 75 and estimate the value of r where the graphs intersect. The graphs intersect where r. cm.., so the radius of the steel ball bearing is about Reflect 7. Discussion Wh is it important to plot multiple points on the graph of the volume function. Because ou are using the graph to estimate the value of the independent variable from a given value of the dependent variable, the graph should be as accurate as possible. Houghton Mifflin Harcourt Publishing Compan Module Lesson Graphing Cubic Functions

9 ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Before students graph an cubic function, encourage them to predict how the graph will look based on the leading coefficient of when the function is written in general form g () = a ( b ( - h) ) + k, and on the values of h and k. SUMMARIZE THE LESSON How do ou use reference points to graph or write equations for combined transformations of f () =? Use the reference points (-, -), (, ), and (, ). Under a combined transformation (-, -) becomes (-b + h, -a + k), (, ) becomes (h, k), and (, ) becomes (b + h, a + k). To graph the combined transformation, ou appl the transformations to the reference points and use those transformed points to graph the transformed function. To write an equation, ou use the transformations of the reference points to solve for a, b, h, and k and use those values to write the equation of the transformed function. Houghton Mifflin Harcourt Publishing Compan Your Turn Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate. 8. Polstrene beads fill a cube-shaped bo with an effective densit of.7 kg/c m (which accounts for the space between the beads). The filled bo weighs kilograms while the empt bo had weighed.5 kilograms. Estimate the inner edge length of the bo. Let l represent the length (in centimeters) of an inner edge of the bo. Since the bo is a cube, the volume V (in cubic centimeters) is V (l) = l. The mass m (in kilograms) of the polstrene beads in the bo is m (l) =.7 V (l) =.7 l. Make a table of values for this function. Length (cm) Elaborate Mass (kg) Mass (kg) m 5 5 Length (cm) Draw the graph of the mass function, recognizing that the graph is a vertical compression of the graph of the parent cubic function b a factor of.7. Then draw the horizontal line m = -.5 =.5 and estimate the value of l where the graphs intersect. The graphs intersect where l 8, so the inner edge length of the bo is about 8 cm. 9. Identif which transformations (stretches or compressions, reflections, and translations) of ƒ () = change the following attributes of the function. a. End behavior Reflections across the -ais (a < ) and reflections across the -ais (b < ) change the end behavior. b. Location of the point of smmetr Vertical translations (k ) and horizontal translations (h ) change the location of the point of smmetr. c. Smmetr about a point No transformations change the function s smmetr about a point.. Essential Question Check-In Describe the transformations ou must perform on the graph of ƒ () = to obtain the graph of g () = a ( - h) + k. If a <, reflect the parent graph across the -ais. Then either stretch the graph verticall b a factor of a if a > or compress the graph verticall b a factor of a if a <. Finall, translate the graph h units horizontall and k units verticall. Module Lesson l Lesson.

10 Evaluate: Homework and Practice EVALUATE. Graph the parent cubic function ƒ () = and use the graph to answer each question. a. State the function s domain and range. The domain is R, and the range is R. b. Identif the function s end behavior. As +, f () +. As -, f () -. c. Identif the graph s - and -intercepts. The graph s onl -intercept is. The graph s onl -intercept is. d. Identif the intervals where the function has positive values and where it has negative values. The function has positive values on the interval (, + ) and negative values on the interval (-, ). e. Identif the intervals where the function is increasing and where it is decreasing. The function is increasing throughout its domain. The function never decreases Online Homework Hints and Help Etra Practice ASSIGNMENT GUIDE Concepts and Skills Eplore Graphing and Analzing f () = Eplore Predicting Transformations of the Graph of f () = Eample Graphing Combined Transformations of f () = Practice Eercise Eercises 9 Eercises 7 f. Tell whether the function is even, odd, or neither. Eplain. The function is odd because f (-) = (-) = - = -f (). Eample Writing Equations for Combined Transformations of f () = Eercises 8 g. Describe the graph s smmetr. The graph is smmetric about the origin. Eample Modeling with a Transformation of f () = Eercises Describe how the graph of g() is related to the graph of ƒ () =.. g () = ( - ) translation of the graph of f () right units.. g () = -5. g () = + 5. g () = () translation of the graph of f () up units. vertical stretch of the graph of f () b a factor of 5 and a reflection across the -ais. horizontal compression of the graph of f () b a factor of. Houghton Mifflin Harcourt Publishing Compan INTEGRATE TECHNOLOGY When graphing a cubic function or an other function on a graphing calculator, students should choose a good viewing window: one that displas the important characteristics of the graph. As a class, have students brainstorm what should show on the graphing screen. Sample answer: the end behavior; the turning point, or point of smmetr of the graph; the -ais of the graph Module Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information.F Analze relationships Skills/Concepts.F Analze relationships Strategic Thinking.A Everda life 5 Skills/Concepts.F Analze relationships Strategic Thinking.G Eplain and justif arguments Graphing Cubic Functions

11 AVOID COMMON ERRORS Students ma be confused about where to start when finding the equation for a combined transformation of the graph of f() =. Eplain that the should compare the corresponding points of the graph to the reference points of the graph of f() =. If the graph is stretched or compressed horizontall, then the ma use the form f () = ( ( b - h) ) + k. If the graph is stretched or compressed verticall, then the ma use the form f () = a ( - h) + k.. g () = ( + ) translation of the graph of f () left unit g () = 8. g () = - 9. g () = ( - _ ) translation of the graph of f () down units. Identif the transformations of the graph of ƒ () = 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 (-, -), (, ), and (, ).. g () = ( ). g () = vertical compression of the graph of f () b a factor of. horizontal stretch of the graph of f () b a factor of _ as well as a reflection across the -ais Houghton Mifflin Harcourt Publishing Compan a horizontal stretch b a factor of. Reference points on the graph of g() are (-, -), (, ), and (, ).. g () = ( - ) -. g () = ( + ) a vertical compression b a factor of. Reference points on the graph of g() are (-, ), (, ) and (, - ). translation of units to the right and units down. Reference points on the graph of g() are (, -), (, -), and (5, -). a translation of unit to the left and units up. Reference points on the graph of g() are (-, ), (-, ), and (, ). Module 5 Lesson A_MTXESE59_UML.indd :5 AM 5 Lesson.

12 . g () = g () = (-( - ) ) a vertical stretch b a factor of a reflection across the -ais a translation of units up Reference points on the graph of g() are (-, 5), (, ), and (, ) a horizontal compression b a factor of a reflection across the -ais a translation of units to the right Reference points on the graph of g() are (, ), (, ), and (, - ).. g () = - ( + ) - 7. g () = ( - ( - ) ) + INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Suggest that students make a graphic organizer similar to the one below to help them remember the differences in the tpes of transformations of f() =. Suggest students choose one or more cubic functions from this eercise set and give an eample of each tpe of transformation. Transformation Vertical shift Horizontal shift Vertical stretch Horizontal compression Eample - - a vertical stretch b a factor of a reflection across the -ais a translation of unit to the left and unit down Reference points on the graph of g() are (-, ), (-, -), and (, -). a horizontal stretch b a factor of a reflection across the ais a translation of unit to the right and units up Reference points on the graph of g() are (-, ), (, ), and (5, ). Houghton Mifflin Harcourt Publishing Compan Module Lesson A_MTXESE59_UML.indd 5-- :5 AM Graphing Cubic Functions

13 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Point out that another name for the point of smmetr of a cubic function in this lesson is the inflection point. This is the point for which the curvature of the graph changes. For a cubic equation with a positive leading coefficient, the rate of change in the -values of the graph decreases as approaches this point from below and then begins to increase again as increases past the point. In later mathematics classes, students will find that the slope of the curve at this point is. 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. 8. g () = ( b ( - h) ) + k (-, -) - - (-5, -) - (, -) (h, k) = (-, -), so h = - and k = -. (b + h, + k) = (b -, - ) = (, -), so b =. g () = ( ( + ) ) - 9. g () = a ( - h) + k (, 7) (, ) (, ) - (h, k) = (, ), so h = and k =. (h +, a + k) = ( +, a + ) = (, ), so a = -. g () = - ( - ) +. g () = ( b ( - h) ) + k - (, -) - - (.5, ) (.5, -). g () = a ( - h) + k (-, ) (-,.5) (,.5) Houghton Mifflin Harcourt Publishing Compan (h, k) = (, -), so h = and k = -. (b + h, + k) = (b +, - ) = (.5, ), so b = -.5. g () = ( -.5 ( - ) ) - = (- ( - ) ) - (h, k) = (-, ), so h = - and k =. (h +, a + k) = (- +, a + ) = (,.5), so a =.5. g () =.5 ( + ) + Module 7 Lesson A_MTXESE59_UML.indd :5 AM 7 Lesson.

14 Use a cubic function to model the situation, and graph the function using calculated values of the function. Then use the graph to obtain the indicated estimate.. Estimate the edge length of a cube of gold with a mass of kg. The densit of gold is.9 kg/ cm. The volume V (in cubic centimeters) of the cube is V (l) = l. The mass m (in kilograms) of the cube is m (l) =.9 V (l) =.9 l. Length (cm) Mass (kg) The graphs intersect where l.75, so the edge length of the cube of gold is about.75 centimeters.. A proposed design for a habitable Mars colon is a hemispherical biodome used to maintain a breathable atmosphere for the colonists. Estimate the radius of the biodome if it is required to contain 5.5 billion cubic feet of air. Mass (kg).5.5 m Length (cm) l INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling For modeling problems involving geometr, review the formulas for the volumes of a sphere and rectangular prism. Students should recall that the volume V of a sphere can be found with the formula V = π r, where r is the radius, and the volume of a rectangular prism is V = lwh, where l is the length of the rectangular base, w is the width of the base, and h is the height of the prism. Since the biodome is a hemisphere, the volume V (in billions of cubic feet) of the biodome is V (r) = π r ).9 r. Radius (thousands of feet) ( _ Volume (billions of cubic feet) The graphs intersect where r., so the radius of the biodome is about. thousand () ft. Volume (billions of cubic feet) V.5.5 Radius (thousands of feet) r Houghton Mifflin Harcourt Publishing Compan Module 8 Lesson Graphing Cubic Functions 8

15 CONNECT VOCABULARY To help students remember how the words shift, stretch, and compression are used in the contet of this lesson, remind students that a shift is a translation, a transformation that does not change the shape or size of the original figure as the figure moves on the coordinate plane. A stretch (or compression), on the other hand, represents what happens to the graph of the parent figure if the leading coefficient of is not equal to one. Reference points from the transformed graph will have more space between the corresponding -values ( stretched ) than the parent graph, or less space between the -values ( compressed ) than points on the parent graph. You can see these changes on the graph for small values of. PEER-TO-PEER DISCUSSION Have students work in pairs. Instruct one student in each pair to sketch the graph of a cubic function of the form f() = a ( - h) + k, while the other gives verbal instructions for each step. Then have the student who sketched the graph write the steps that were followed. Have students switch roles and repeat the eercise using a cubic function of the form f() = b ( - h ) + k. JOURNAL Have students compare and contrast algebraic representations of vertical and horizontal transformations of cubic functions. Houghton Mifflin Harcourt Publishing Compan. Multiple Response Select the transformations of the graph of the parent cubic function that result in the graph of g () = ( ( - ) ) +. A. Horizontal stretch b a factor of E. Translation unit up B. Horizontal compression b a factor of F. Translation unit down C. Vertical stretch b a factor of G. Translation units left D. Vertical compression b a factor of H. Translation units right H.O.T. Focus on Higher Order Thinking 5. Justif Reasoning Eplain how horizontall stretching (or compressing) the graph of ƒ () = b a factor of b can be equivalent to verticall compressing (or stretching) the graph of ƒ () = b a factor of a. Horizontall stretching or compressing the graph of the parent cubic function b a factor of b results in the graph of the function g () = ( b ). You can rewrite g() as follows: g () = ( b ) = (. Critique Reasoning A student reasoned that g () = ( - h) can be rewritten as g () = - h, so a horizontal translation of h units is equivalent to a vertical translation of - h units. Is the student correct? Eplain. b) = b The function is now in the form g () = a where a =. Notice that when b >, a <, b which means that a horizontal stretch is equivalent to a vertical compression. Similarl, when b <, a >, which means that a horizontal compression is equivalent to a vertical stretch. No, the student isn t correct. There is no propert of eponents that allows ou to distribute an eponent to a sum or difference, so g () = ( - h) cannot be rewritten in the form g () = + k. From a geometric perspective, it doesn t make sense to think that there can be a vertical translation of a graph that is equivalent to a horizontal translation of the graph ecept in the trivial case where the graph is not moved at all (h = k = ). Module 9 Lesson 9 Lesson.

16 Lesson Performance Task Julio wants to purchase a spherical aquarium and fill it with salt water, which has an average densit of.7 g/c m. He has found a compan that sells four sizes of spherical aquariums. Aquarium Size Diameter (cm) Small 5 Medium Large 5 Etra large a. If the stand for Julio s aquarium will support a maimum of 5 kg, what is the largest size tank that he should bu? Eplain our reasoning. b. Julio s friend suggests that he could bu a larger tank if he uses fresh water, which has a densit of. g/c m. Do ou agree with the friend? Wh or wh not? a. The volume V (in cubic centimeters) of a sphere with diameter d (in centimeters) is V (d) = _ π ( d_ ) = π d. If the sphere is filled with salt water having a densit of.7 g/cm, then the mass m (in kilograms) of the salt water is m (V) =.7 V =.7V. The table summarizes the volume and mass of salt water for each size of tank. Aquarium Size Volume (c m ) Mass of Water (kg) Small,7.8 Medium,.5 Large 7,7 9. Etra large, Some students ma sa that Julio should bu the large tank, because the mass of the water, 9. kg, is less than 5 kg. Other students ma sa that he should bu the medium tank, because he has to consider the mass of the empt tank, which will likel make the total mass greater than 5 kg. b. Students answers will depend on how the answered Part a. If the answered that Julio should bu the large tank, then the minor difference in densit will have no effect on his choice. If the said he should bu the medium tank, some students ma decide that the lesser densit of fresh water might reduce the mass of the water just enough to allow him to bu the large tank. Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS Make sure students are multipling the volume b the densit instead of dividing b the densit. Have students write the units net to the values as the do the calculation. If the units cancel properl, the final units should be mass. Ask students what the real-life consequence of dividing b the densit would be. QUESTIONING STRATEGIES Wh would using fresh water allow Julio to bu a larger tank? Fresh water has a lower densit and therefore has a lower mass for the same volume. If the diameter of the Medium tank is twice the diameter of the Small tank, wh isn t the volume of the Medium tank twice that of the Small tank? The volume is described b a cubic function. The ratio of the volumes is the ratio of their diameters cubed. Wh is it important to know the densit of the water? The densit is the mass per volume. If the volume is known, then the densit will give the mass that the volume will hold. Module Lesson EXTENSION ACTIVITY Julio decides he wants to fill one-third of his aquarium with sand that has a densit of. g/c m. The remaining two-thirds of the aquarium will be filled with water. Have students answer the questions in the Performance Task using this new information. Then have them graph the mass in the aquarium compared with the aquarium size, both with and without the sand. Ask them how the addition of the sand affects the total mass in the aquarium and if this information will help Julio decide between a Medium and a Large tank. Scoring Rubric points: The student s answer is an accurate and complete eecution of the task or tasks. point: The student s answer contains attributes of an appropriate response but is flawed. points: The student s answer contains no attributes of an appropriate response. Graphing Cubic Functions