Graphing f ( x) = ax 2 + c

Transcription

1 . Graphing f ( ) = a + c Essential Question How does the value of c affect the graph of f () = a + c? Graphing = a + c Work with a partner. Sketch the graphs of the functions in the same coordinate plane. What do ou notice? a. f () = and g() = + b. f () = and g() = 0 0 Finding -Intercepts of Graphs Work with a partner. Graph each function. Find the -intercepts of the graph. Eplain how ou found the -intercepts. a. = 7 b. = + USING TOOLS STRATEGICALLY To be proficient in math, ou need to consider the available tools, such as a graphing calculator, when solving a mathematical problem. Communicate Your Answer 3. How does the value of c affect the graph of f () = a + c?. Use a graphing calculator to verif our answers to Question The figure shows the graph of a quadratic function of the form = a + c. Describe possible values of a and c. Eplain our reasoning. Section. Graphing f() = a + c 5

2 . Lesson What You Will Learn Core Vocabular zero of a function, p. Previous translation verte of a parabola ais of smmetr vertical stretch vertical shrink Graph quadratic functions of the form f () = a + c. Solve real-life problems involving functions of the form f () = a + c. Graphing f ( ) = a + c Core Concept Graphing f ( ) = a + c When c > 0, the graph of f () = a + c is a vertical translation c units up of the graph of f () = a. c > 0 c = 0 When c < 0, the graph of f () = a + c is a vertical translation c units down of the graph of f () = a. The verte of the graph of f () = a + c is (0, c), and the ais of smmetr is = 0. c < 0 Graphing = + c Graph g() =. Compare the graph to the graph of f () =. Step Make a table of values. 0 g() Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. REMEMBER The graph of = f () + k is a vertical translation, and the graph of = f ( h) is a horizontal translation of the graph of f. g() = 3 f() = Both graphs open up and have the same ais of smmetr, = 0. The verte of the graph of g, (0, ), is below the verte of the graph of f, (0, 0), because the graph of g is a vertical translation units down of the graph of f. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of f () =.. g() = 5. h() = + 3 Chapter Graphing Quadratic Functions

3 Graphing = a + c Graph g() = +. Compare the graph to the graph of f () =. Step Make a table of values. g() = + f() = 0 g() Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. Both graphs open up and have the same ais of smmetr, = 0. The graph of g is narrower, and its verte, (0, ), is above the verte of the graph of f, (0, 0). So, the graph of g is a vertical stretch b a factor of and a vertical translation unit up of the graph of f. Translating the Graph of = a + c Let f () = and g() = f () 7. a. Describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. b. Write an equation that represents g in terms of. a. The function g is of the form = f () + k, where k = 7. So, the graph of g is a vertical translation 7 units down of the graph of f. 0 f () 0 0 g() f ( ) 7 b. g() = f () 7 Write the function g. g f = Substitute for f (). = Subtract. So, the equation g() = represents g in terms of. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of f () =. 3. g() = 5. h() = + 5. Let f () = 3 and g() = f () + 3. a. Describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. b. Write an equation that represents g in terms of. Section. Graphing f() = a + c 7

4 Solving Real-Life Problems A zero of a function f is an -value for which f () = 0. A zero of a function is an -intercept of the graph of the function. Solving a Real-Life Problem The function f (t) = t + s 0 represents the approimate height (in feet) of a falling object t seconds after it is dropped from an initial height s 0 (in feet). An egg is dropped from a height of feet. a. After how man seconds does the egg hit the ground? ft b. Suppose the initial height is adjusted b k feet. How will this affect part (a)?. Understand the Problem You know the function that models the height of a falling object and the initial height of an egg. You are asked to find how man seconds it takes the egg to hit the ground when dropped from the initial height. Then ou need to describe how a change in the initial height affects how long it takes the egg to hit the ground. COMMON ERROR The graph in Step shows the height of the object over time, not the path of the object.. Make a Plan Use the initial height to write a function that models the height of the egg. Use a table to graph the function. Find the zero(s) of the function to answer the question. Then eplain how vertical translations of the graph affect the zero(s) of the function. 3. Solve the Problem a. The initial height is feet. So, the function f (t) = t + represents the height of the egg t seconds after it is dropped. The egg hits the ground when f (t) = 0. Step Make a table of values and f(t) = t sketch the graph. + t 0 f (t) 0 3 Step Find the positive zero of the function. When t =, f (t) = 0. So, the zero is. The egg hits the ground seconds after it is dropped. 3 t b. When the initial height is adjusted b k feet, the graph of f is translated up k units when k > 0 or down k units when k < 0. So, the -intercept of the graph of f will move right when k > 0 or left when k < 0. When k > 0, the egg will take more than seconds to hit the ground. When k < 0, the egg will take less than seconds to hit the ground.. Look Back To check that the egg hits the ground seconds after it is dropped, ou can solve 0 = t + b factoring. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Eplain wh onl nonnegative values of t are used in Eample. 7. WHAT IF? The egg is dropped from a height of 00 feet. After how man seconds does the egg hit the ground? Chapter Graphing Quadratic Functions

5 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY State the verte and ais of smmetr of the graph of = a + c.. WRITING How does the graph of = a + c compare to the graph of = a? Monitoring Progress and Modeling with Mathematics In Eercises 3, graph the function. Compare the graph to the graph of f() =. (See Eample.) 3. g() = +. h() = + 5. p() = 3. q() = In Eercises 7, graph the function. Compare the graph to the graph of f() =. (See Eample.) 7. g() = + 3. h() = 7 9. s() = 0. t() = 3 +. p() = 3. q() = + In Eercises 3, describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. Write an equation that represents g in terms of. (See Eample 3.) 3. f () = 3 +. f () = + g() = f () + g() = f () 5. f () =. f () = 5 g() = f () 3 g() = f () ERROR ANALYSIS Describe and correct the error in comparing the graphs. = 3 + = The graph of = 3 + is a vertical shrink b a factor of 3 and a translation units up of the graph of =.. ERROR ANALYSIS Describe and correct the error in graphing and comparing f () = and g() = Both graphs open up and have the same ais of smmetr. However, the verte of the graph of g, (0, 0), is 0 units above the verte of the graph of f, (0, 0). In Eercises 9, find the zeros of the function. 9. = 0. = 3. f () = + 5. f () = f () =. f () = f () = + 3. f () = MODELING WITH MATHEMATICS A water balloon is dropped from a height of feet. (See Eample.) a. After how man seconds does the water balloon hit the ground? b. Suppose the initial height is adjusted b k feet. How does this affect part (a)?. MODELING WITH MATHEMATICS The function = + 3 represents the height (in feet) of an apple seconds after falling from a tree. Find and interpret the - and -intercepts. g f Section. Graphing f() = a + c 9

6 In Eercises 9 3, sketch a parabola with the given characteristics. 9. The parabola opens up, and the verte is (0, 3). 30. The verte is (0, ), and one of the -intercepts is. 3. The related function is increasing when < 0, and the zeros are and. 3. The highest point on the parabola is (0, 5). 33. DRAWING CONCLUSIONS You and our friend both drop a ball at the same time. The function h() = + 5 represents the height (in feet) of our ball after seconds. The function g() = represents the height (in feet) of our friend s ball after seconds. a. Write the function T() = h() g(). What does T() represent? b. When our ball hits the ground, what is the height of our friend s ball? Use a graph to justif our answer. 3. MAKING AN ARGUMENT Your friend claims that in the equation = a + c, the verte changes when the value of a changes. Is our friend correct? Eplain our reasoning. 35. MATHEMATICAL CONNECTIONS The area A (in square feet) of a square patio is represented b A =, where is the length of one side of the patio. You add square feet to the patio, resulting in a total area of 9 square feet. What are the dimensions of the original patio? Use a graph to justif our answer. 3. HOW DO YOU SEE IT? The graph of f () = a + c is shown. Points A and B are the same distance from the verte of the graph of f. Which point is closer to the verte of the graph of f as c increases? A B Maintaining Mathematical Proficienc f Evaluate the epression when a = and b = 3. (Skills Review Handbook). a b 3. b a. 37. REASONING Describe two methods ou can use to find the zeros of the function f (t) = t Check our answer b graphing. 3. PROBLEM SOLVING The paths of water from three different garden waterfalls are given below. Each function gives the height h (in feet) and the horizontal distance d (in feet) of the water. Waterfall h = 3.d +. Waterfall h = 3.5d +.9 Waterfall 3 h =.d +. a. Which waterfall drops water from the highest point? b. Which waterfall follows the narrowest path? c. Which waterfall sends water the farthest? 39. WRITING EQUATIONS Two acorns fall to the ground from an oak tree. One falls 5 feet, while the other falls 3 feet. a. For each acorn, write an equation that represents the height h (in feet) as a function of the time t (in seconds). b. Describe how the graphs of the two equations are related. 0. THOUGHT PROVOKING One of two classic problems in calculus is to find the area under a curve. Approimate the area of the region bounded b the parabola and the -ais. (, 0) Show our work. (0, ) (, 0). CRITICAL THINKING A cross section of the parabolic surface of the antenna shown 30 0 can be modeled b = 0.0, where and are measured 0 0 in feet. The antenna is moved up so that the outer edges of the dish are 5 feet above the -ais. Where is the verte of the cross section located? Eplain. Reviewing what ou learned in previous grades and lessons a b 3a + b 5. b + a ab Chapter Graphing Quadratic Functions