1.1. Parent Functions and Transformations Essential Question What are the characteristics of some of the basic parent functions?
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1 1.1 Parent Functions and Transformations Essential Question What are the characteristics of some of the basic parent functions? Identifing Basic Parent Functions JUSTIFYING CONCLUSIONS To be proficient in math, ou need to justif our conclusions and communicate them clearl to others. Work with a partner. Graphs of eight basic parent functions are shown below. Classif each function as constant, linear, absolute value, quadratic, square root, cubic, reciprocal, or eponential. Justif our reasoning. a. b. c. d. e. f. g. h. Communicate Your Answer. What are the characteristics of some of the basic parent functions? 3. Write an equation for each function whose graph is shown in Eploration 1. Then use a graphing calculator to verif that our equations are correct. Section 1.1 Parent Functions and Transformations 3
2 1.1 Lesson What You Will Learn Core Vocabular parent function, p. transformation, p. 5 translation, p. 5 reflection, p. 5 vertical stretch, p. vertical shrink, p. Previous function domain range slope scatter plot Identif families of functions. Describe transformations of parent functions. Describe combinations of transformations. Identifing Function Families Functions that belong to the same famil share ke characteristics. The parent function is the most basic function in a famil. Functions in the same famil are transformations of their parent function. Core Concept Parent Functions Famil Constant Linear Absolute Value Quadratic Rule f() = 1 f() = f() = f() = Graph Domain All real numbers All real numbers All real numbers All real numbers Range = 1 All real numbers 0 0 LOOKING FOR STRUCTURE You can also use function rules to identif functions. The onl variable term in f is an -term, so it is an absolute value function. Identifing a Function Famil Identif the function famil to which f belongs. Compare the graph of f to the graph of its parent function. The graph of f is V-shaped, so f is an absolute value function. The graph is shifted up and is narrower than the graph of the parent absolute value function. The domain of each function is all real numbers, but the range of f is 1 and the range of the parent absolute value function is 0. f() = + 1 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Identif the function famil to which g belongs. Compare the graph of g to the graph of its parent function. 1 g() = ( 3) Chapter 1 Linear Functions
3 Describing Transformations A transformation changes the size, shape, position, or orientation of a graph. A translation is a transformation that shifts a graph horizontall and/or verticall but does not change its size, shape, or orientation. REMEMBER The slope-intercept form of a linear equation is = m + b, where m is the slope and b is the -intercept. Graphing and Describing Translations Graph g() = and its parent function. Then describe the transformation. The function g is a linear function with a slope of 1 and a -intercept of. So, draw a line through the point (0, ) with a slope of 1. The graph of g is units below the graph of the parent linear function f. f() = So, the graph of g() = is a vertical translation units down of the graph of the parent linear function. (0, ) g() = A reflection is a transformation that flips a graph over a line called the line of refl ection. A reflected point is the same distance from the line of reflection as the original point but on the opposite side of the line. REMEMBER The function p() = is written in function notation, where p() is another name for. Graphing and Describing Reflections Graph p() = and its parent function. Then describe the transformation. The function p is a quadratic function. Use a table of values to graph each function. = = f() = p() = The graph of p is the graph of the parent function flipped over the -ais. So, p() = is a reflection in the -ais of the parent quadratic function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function and its parent function. Then describe the transformation.. g() = h() = ( ). n() = Section 1.1 Parent Functions and Transformations 5
4 Another wa to transform the graph of a function is to multipl all of the -coordinates b the same positive factor (other than 1). When the factor is greater than 1, the transformation is a vertical stretch. When the factor is greater than 0 and less than 1, it is a vertical shrink. Graphing and Describing Stretches and Shrinks Graph each function and its parent function. Then describe the transformation. a. g() = b. h() = 1 REASONING ABSTRACTLY To visualize a vertical stretch, imagine pulling the points awa from the -ais. To visualize a vertical shrink, imagine pushing the points toward the -ais. a. The function g is an absolute value function. Use a table of values to graph the functions. g() = = = The -coordinate of each point on g is two times the -coordinate of the corresponding point on the parent function. f() = So, the graph of g() = is a vertical stretch of the graph of the parent absolute value function. b. The function h is a quadratic function. Use a table of values to graph the functions. = = 1 f() = h() = Chapter 1 Linear Functions The -coordinate of each point on h is one-half of the -coordinate of the corresponding point on the parent function. So, the graph of h() = 1 is a vertical shrink of the graph of the parent quadratic function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function and its parent function. Then describe the transformation. 5. g() = 3. h() = 3 7. c() = 0.
5 Combinations of Transformations You can use more than one transformation to change the graph of a function. Describing Combinations of Transformations Use a graphing calculator to graph g() = and its parent function. Then describe the transformations. 8 The function g is an absolute value function. f The graph shows that g() = is a reflection in the -ais followed b a translation 5 units left and 3 units down of the graph of the parent absolute value function. 1 g Modeling with Mathematics Time (seconds), Height (feet), The table shows the height of a dirt bike seconds after jumping off a ramp. What tpe of function can ou use to model the data? Estimate the height after 1.75 seconds. 1. Understand the Problem You are asked to identif the tpe of function that can model the table of values and then to find the height at a specific time.. Make a Plan Create a scatter plot of the data. Then use the relationship shown in the scatter plot to estimate the height after 1.75 seconds. 3. Solve the Problem Create a scatter plot. 30 The data appear to lie on a curve that resembles a quadratic function. Sketch the curve. So, ou can model the data with a quadratic function. The graph shows that the height is about 15 feet after 1.75 seconds Look Back To check that our solution is reasonable, analze the values in the table. Notice that the heights decrease after 1 second. Because 1.75 is between 1.5 and, the height must be between 0 feet and 8 feet. 8 < 15 < Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use a graphing calculator to graph the function and its parent function. Then describe the transformations. 8. h() = d() = 3( 5) The table shows the amount of fuel in a chainsaw over time. What tpe of function can ou use to model the data? When will the tank be empt? Time (minutes), Fuel remaining (fluid ounces), Section 1.1 Parent Functions and Transformations 7
6 1.1 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The function f() = is the of f() = 3.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. What are the vertices of the figure after a reflection in the -ais, followed b a translation units right? What are the vertices of the figure after a translation units up and units right? What are the vertices of the figure after a translation units right, followed b a reflection in the -ais? What are the vertices of the figure after a translation units up, followed b a reflection in the -ais? Monitoring Progress and Modeling with Mathematics In Eercises 3, identif the function famil to which f belongs. Compare the graph of f to the graph of its parent function. (See Eample 1.) 3.. f() = f() = 5 f() = + 3 f() = 3 7. MODELING WITH MATHEMATICS At 8:00 a.m., the temperature is 3 F. The temperature increases F each hour for the net 7 hours. Graph the temperatures over time t (t = 0 represents 8:00 a.m.). What tpe of function can ou use to model the data? Eplain. 8. MODELING WITH MATHEMATICS You purchase a car from a dealership for $10,000. The trade-in value of the car each ear after the purchase is given b the function f() = 10, What tpe of function models the trade-in value? In Eercises 9 18, graph the function and its parent function. Then describe the transformation. (See Eamples and 3.) 9. g() = f() = 11. f() = 1 1. h() = ( + ) 13. g() = 5 1. f() = h() = 1. g() = 17. f() = f() = 8 Chapter 1 Linear Functions
7 In Eercises 19, graph the function and its parent function. Then describe the transformation. (See Eample.) 19. f() = 1 0. g() = 3 1. f() =. h() = h() = 3. g() = 3 5. h() = 3. f() = 1 In Eercises 7 3, use a graphing calculator to graph the function and its parent function. Then describe the transformations. (See Eample 5.) 7. f() = h() = h() = f() = g() = 1 3. f() = f() = ( + 3) g() = 1 1 ERROR ANALYSIS In Eercises 35 and 3, identif and correct the error in describing the transformation of the parent function The graph is a reflection in the -ais and a vertical shrink of the parent quadratic function. The graph is a translation 3 units right of the parent absolute value function, so the function is f() = + 3. MATHEMATICAL CONNECTIONS In Eercises 37 and 38, find the coordinates of the figure after the transformation. 37. Translate units 38. Reflect in the -ais. down. B A C D USING TOOLS In Eercises 39, identif the function famil and describe the domain and range. Use a graphing calculator to verif our answer. 39. g() = h() = 3 + A C 1. g() = 3 +. f() = f() = 5. f() = + B 5. MODELING WITH MATHEMATICS The table shows the speeds of a car as it travels through an intersection with a stop sign. What tpe of function can ou use to model the data? Estimate the speed of the car when it is 0 ards past the intersection. (See Eample.) Displacement from sign (ards), Speed (miles per hour), THOUGHT PROVOKING In the same coordinate plane, sketch the graph of the parent quadratic function and the graph of a quadratic function that has no -intercepts. Describe the transformation(s) of the parent function. 7. USING STRUCTURE Graph the functions f() = and g() =. Are the equivalent? Eplain. Section 1.1 Parent Functions and Transformations 9
8 8. HOW DO YOU SEE IT? Consider the graphs of f, g, and h. h f g a. Does the graph of g represent a vertical stretch or a vertical shrink of the graph of f? Eplain our reasoning. b. Describe how to transform the graph of f to obtain the graph of h. 9. MAKING AN ARGUMENT Your friend sas two different translations of the graph of the parent linear function can result in the graph of f() =. Is our friend correct? Eplain. 50. DRAWING CONCLUSIONS A person swims at a constant speed of 1 meter per second. What tpe of function can be used to model the distance the swimmer travels? If the person has a 10-meter head start, what tpe of transformation does this represent? Eplain. 51. PROBLEM SOLVING You are plaing basketball with our friends. The height (in feet) of the ball above the ground t seconds after a shot is released from our hand is modeled b the function f(t) = 1t + 3t a. Without graphing, identif the tpe of function that models the height of the basketball. b. What is the value of t when the ball is released from our hand? Eplain our reasoning. c. How man feet above the ground is the ball when it is released from our hand? Eplain. 5. MODELING WITH MATHEMATICS The table shows the batter lives of a computer over time. What tpe of function can ou use to model the data? Interpret the meaning of the -intercept in this situation. Time (hours), Batter life remaining, 1 80% 3 0% 5 0% 0% 8 0% 53. REASONING Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Eplain our reasoning. a. f() = 3 b. f() = ( 8) c. f() = + + d. f() = 5. CRITICAL THINKING Use the values 1, 0, 1, and in the correct bo so the graph of each function intersects the -ais. Eplain our reasoning. a. f() = b. f() = c. f() = + 1 d. f() = Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Determine whether the ordered pair is a solution of the equation. (Skills Review Handbook) 55. f() = + ; (1, 3) 5. f() = 3; (, 5) 57. f() = 3; (5, ) 58. f() = ; (1, 8) Find the -intercept and the -intercept of the graph of the equation. (Skills Review Handbook) 59. = 0. = = 1. = 8 10 Chapter 1 Linear Functions
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