Name Class Date - Transforming Quadratic Functions Going Deeper Essential question: How can ou obtain the graph of g() = a( h ) + k from the graph of f () =? 1 F-BF..3 ENGAGE Understanding How to Graph g() = a( - h ) + k The sequence of graphs below shows how ou can obtain the graph of g() = ( - 3) + 1 from the graph of the parent quadratic function f () = using transformations. 1. Start with the graph of =.. Stretch the graph verticall b a factor of to obtain the graph of =. 3. Translate the graph of = right 3 units and up 1 unit to obtain the graph of = ( - 3 ) + 1. - - - - - - - - - Houghton Mifflin Harcourt Publishing Compan REFLECT 1a. The verte of the graph of f () = is, while the verte of the graph of g() = ( - 3 ) + 1 is. 1b. If ou start at the verte of the graph of f () = and move 1 unit to the right or left, how must ou move verticall to get back to the graph? 1c. If ou start at the verte of the graph of g() = ( - 3 ) + 1 and move 1 unit to the right or left, how must ou move verticall to get back to the graph? 1d. Based on our answers to Questions 1a c, describe how ou could graph g() = a( - h) + k directl, without using transformed graphs. Chapter 31 Lesson
F-BF..3 EXAMPLE Graphing g() = a( - h ) + k Graph g() = -3( + 1 ) -. A Identif and plot the verte. Verte: B Identif and plot other points based on the fact that f (±1) = 1 for the parent function f () =. If ou move 1 unit right or left from the verte in part A, how must ou move verticall to be on the graph of g()? What points are ou at? C Use the plotted points to draw a parabola. REFLECT a. List the transformations of the graph of the parent function f () =, in the order that ou would perform them, to obtain the graph of g() = -3( + 1 ) -. b. Before graphing g() = -3( + 1 ) -, would ou have epected the graph to open up or down? Wh? c. Suppose ou changed the -3 in g() = -3( + 1 ) - to -. Which of the points that ou identified in parts A and B of the eample would change? What coordinates would the now have? Houghton Mifflin Harcourt Publishing Compan Chapter 3 Lesson
3 F-BF.1.1 EXAMPLE Writing a Quadratic Function from a Graph A house painter standing on a ladder drops a paintbrush, which falls to the ground. The paintbrush s height above the ground (in feet) is given b a function of the form f (t) = a(t - h ) + k where t is the time (in seconds) since the paintbrush was dropped. Because f (t) is a quadratic function, its graph is a parabola. Onl the portion of the parabola that lies in Quadrant I and on the aes is shown because onl nonnegative values of t and f (t) make sense in this situation. The verte of the parabola lies on the vertical ais. Use the graph to find an equation for f (t). A The verte of the parabola is (h, k) = (, ). Height (feet) 3 1 1 1 1 1 (1, 1).5 1.5.5 Time (seconds) Substitute the values of h and k into the general equation for f (t) to get f (t) = a ( t - ) +. B From the graph ou can see that f (1) =. Substitute 1 for t and for f (t) to determine the value of a for this function: = a ( 1 - ) + = a C Write the equation for the function: f (t) = Houghton Mifflin Harcourt Publishing Compan REFLECT 3a. Using the graph, estimate how much time elapses until the paintbrush hits the ground: t 3b. Using the value of t from Question 3a and the equation for the height function from part C of the eample, find the value of f (t). How does this help ou check the reasonableness of the equation? Chapter 33 Lesson
PRACTICE Graph each quadratic function. 1. f () = ( - ) + 3. f () = -( - 1 ) + 3. f () = 1 ( - ). f () = - 1 3-3 5. A roofer working on a roof accidentall drops a hammer, which falls to the ground. The hammer s height above the ground (in feet) is given b a function of the form f (t) = a(t - h ) + k where t is the time (in seconds) since the hammer was dropped. Because f (t) is a quadratic function, its graph is a parabola. Onl the portion of the parabola that lies in Quadrant I and on the aes is shown because onl nonnegative values of t and f (t) make sense in this situation. The verte of the parabola lies on the vertical ais. a. Use the graph to find an equation for f (t). b. Eplain how ou can use the graph s t-intercept to check the reasonableness of our equation. Height (feet) 5 5 35 3 5 15 1 5 (1, 9).5 1.5.5 Time (seconds) Houghton Mifflin Harcourt Publishing Compan Chapter 3 Lesson
Name Class Date - Additional Practice 1. ( ) = 3 ; ( ) =. ( ) = 1 ; ( ) = 5 ; ( ) = 3. ( ) = ; ( ) = 3 ; ( ) = 1. ( ) =.5 ; ( ) = 1 ; ( ) = 1 3 = 5. ( ) = 5 + 1. ( ) = 1 3 7. ( ) = 3 +. ( ) = 3 + 1 Houghton Mifflin Harcourt Publishing Compan 9. Two sandbags are dropped from a hot air balloon, one from a height of feet and the other from a height of 1 feet. a. Write the two height functions. 1 ( ) = ( ) = b. Sketch and compare their graphs. c. Tell when each sandbag reaches the ground. Chapter 35 Lesson
Problem Solving Write the correct answer. 1. Two construction workers working at different heights on a skscraper dropped their hammers at the same time. The first was working at a height of ft, the second at a height of 1 ft. Write the two functions that describe the heights of the hammers. 3. Based on the graphs ou drew in problem, how long will it take each hammer to reach the ground?. Graph the two functions ou found in problem 1 on the grid below. The pull of gravit varies from planet to planet. The graph shows the height of objects dropped from 5 ft on the surface of four planets. Use this graph to answer questions. Select the best answer. 5. Which of the graphs represents an object dropped on Earth? A Graph 1 C Graph 3 B Graph D Graph. Of the four planets, Jupiter has the strongest gravit. Which of the four graphs represents the height of the object dropped on Jupiter? F Graph 1 H Graph 3 G Graph J Graph. Due to its small size, Pluto has a ver weak pull of gravit. Which of the equations below represents the graph of the object dropped on Pluto? A h(t) = 1 + 5 B h(t) = 1 + 5 C h(t) = + 5 D h(t) = 1.5 + 5 Houghton Mifflin Harcourt Publishing Compan Chapter 3 Lesson