Graphing f ( x) = ax 2

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1 . Graphing f ( ) = a Essential Question What are some of the characteristics of the graph of a quadratic function of the form f () = a? Graphing Quadratic Functions Work with a partner. Graph each quadratic function. Compare each graph to the graph of f () =. a. g() = 3 b. g() = 5 0 f() = f() = c. g() = 0. d. g() = 0 f() = 0 f() = REASONING QUANTITATIVELY To be proficient in math, ou need to make sense of quantities and their relationships in problem situations. Communicate Your Answer. What are some of the characteristics of the graph of a quadratic function of the form f () = a? 3. How does the value of a affect the graph of f () = a? Consider 0 < a <, a >, < a < 0, and a <. Use a graphing calculator to verif our answers.. The figure shows the graph of a quadratic function of the form = a. Which of the intervals in Question 3 describes the value of a? Eplain our reasoning. 7 Section. Graphing f() = a 9

2 . Lesson What You Will Learn Core Vocabular quadratic function, p. 0 parabola, p. 0 verte, p. 0 ais of smmetr, p. 0 Previous domain range vertical shrink vertical stretch reflection REMEMBER The notation f () is another name for. Identif characteristics of quadratic functions. Graph and use quadratic functions of the form f () = a. Identifing Characteristics of Quadratic Functions A quadratic function is a nonlinear function that can be written in the standard form = a + b + c, where a 0. The U-shaped graph of a quadratic function is called a parabola. In this lesson, ou will graph quadratic functions, where b and c equal 0. Core Concept Characteristics of Quadratic Functions The parent quadratic function is f () =. The graphs of all other quadratic functions are transformations of the graph of the parent quadratic function. The lowest point on a parabola that opens up or the highest point on a parabola that opens down is the verte. The verte of the graph of f () = is (0, 0). decreasing ais of smmetr increasing verte The vertical line that divides the parabola into two smmetric parts is the ais of smmetr. The ais of smmetr passes through the verte. For the graph of f () =, the ais of smmetr is the -ais, or = 0. Identifing Characteristics of a Quadratic Function Consider the graph of the quadratic function. Using the graph, ou can identif characteristics such as the verte, ais of smmetr, and the behavior of the graph, as shown. Function is decreasing. Function is increasing. ais of smmetr: = verte: (, ) You can also determine the following: The domain is all real numbers. The range is all real numbers greater than or equal to. When <, increases as decreases. When >, increases as increases. 0 Chapter Graphing Quadratic Functions

3 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Identif characteristics of the quadratic function and its graph... REMEMBER The graph of = a f () is a vertical stretch or shrink b a factor of a of the graph of = f (). The graph of = f () is a reflection in the -ais of the graph of = f (). Graphing and Using f ( ) = a Core Concept Graphing f ( ) = a When a > 0 When 0 < a <, the graph of f () = a is a vertical shrink of the graph of f () =. When a >, the graph of f () = a is a vertical stretch of the graph of f () =. 7 3 a = a > 0 < a < Graphing f ( ) = a When a < 0 When < a < 0, the graph of f () = a is a vertical shrink with a reflection in the -ais of the graph of f () =. When a <, the graph of f () = a is a vertical stretch with a reflection in the -ais of the graph of f () =. a = < a < 0 a < Graphing = a When a > 0 Graph g() =. Compare the graph to the graph of f () =. SOLUTION Step Make a table of values. 0 g() 0 g() = f() = Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. Both graphs open up and have the same verte, (0, 0), and the same ais of smmetr, = 0. The graph of g is narrower than the graph of f because the graph of g is a vertical stretch b a factor of of the graph of f. Section. Graphing f() = a

4 STUDY TIP To make the calculations easier, choose -values that are multiples of 3. Graphing = a When a < 0 Graph h() = 3. Compare the graph to the graph of f () =. SOLUTION Step Make a table of values h() Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. f() = The graphs have the same verte, (0, 0), and the same ais of smmetr, = 0, but the graph of h opens down and is wider than the graph of f. So, the graph of h is a vertical shrink b a factor of 3 and a reflection in the -ais of the graph of f. h() = 3 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() = 3 5. n() = 3. p() = 3 7. q() = 0.. g() = Solving a Real-Life Problem (, ) = (, ) The diagram at the left shows the cross section of a satellite dish, where and are measured in meters. Find the width and depth of the dish. SOLUTION (0, 0) Use the domain of the function to find the width of the dish. Use the range to find the depth. The leftmost point on the graph is (, ), and the rightmost point is (, ). So, the domain is, which represents meters. width depth The lowest point on the graph is (0, 0), and the highest points on the graph are (, ) and (, ). So, the range is 0, which represents meter. So, the satellite dish is meters wide and meter deep. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 9. The cross section of a spotlight can be modeled b the graph of = 0.5, where and are measured in inches and. Find the width and depth of the spotlight. Chapter Graphing Quadratic Functions

5 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY What is the U-shaped graph of a quadratic function called?. WRITING When does the graph of a quadratic function open up? open down? Monitoring Progress and Modeling with Mathematics In Eercises 3 and, identif characteristics of the quadratic function and its graph. (See Eample.) ERROR ANALYSIS Describe and correct the error in graphing and comparing = and = 0.5. = = 0.5. The graphs have the same verte and the same ais of smmetr. The graph of = 0.5 is narrower than the graph of =.. MODELING WITH MATHEMATICS The arch support of a bridge can be modeled b = 0.00, where and are measured in feet. Find the height and width of the arch. (See Eample.) In Eercises 5, graph the function. Compare the graph to the graph of f () =. (See Eamples and 3.) 5. g() =. b() =.5 7. h() =. j() = m() = 0. q() =. k() = 0.. p() = 9 3 In Eercises 3, use a graphing calculator to graph the function. Compare the graph to the graph of =. 3. =. = = 0.0. = PROBLEM SOLVING The breaking strength z (in pounds) of a manila rope can be modeled b z = 900d, where d is the diameter (in inches) of the rope. a. Describe the domain and range of the function. b. Graph the function using the domain in part (a). c. A manila rope has four times the breaking strength of another manila rope. Does the stronger rope have four times the diameter? Eplain. d Section. Graphing f() = a 3

6 0. HOW DO YOU SEE IT? Describe the possible values of a. a. ABSTRACT REASONING In Eercises 9, determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning.. The graph of f () = a is narrower than the graph of g() = when a > The graph of f () = a is narrower than the graph of g() = when a >. b. g() = a f() = f() = g() = a. The graph of f () = a is wider than the graph of g() = when 0 < a <. 9. The graph of f () = a is wider than the graph of g() = d when a > d. 30. THOUGHT PROVOKING Draw the isosceles triangle shown. Divide each leg into eight congruent segments. Connect the highest point of one leg with the lowest point of the other leg. Then connect the second highest point of one leg to the second lowest point of the other leg. Continue this process. Write a quadratic equation whose graph models the shape that appears. (0, ) ANALYZING GRAPHS In Eercises 3, use the graph. leg leg. When is each function increasing?. When is each function decreasing? f() = a, a > 0 g() = a, a < 0 3. Which function could include the point (, 3)? Find the value of a when the graph passes through (, 3).. REASONING Is the -intercept of the graph of = a alwas 0? Justif our answer. 5. REASONING A parabola opens up and passes through (, ) and (, 3). How do ou know that (, ) is not the verte? (, ) (, ) base 3. MAKING AN ARGUMENT The diagram shows the parabolic cross section of a swirling glass of water, where and are measured in centimeters. a. About how wide is the mouth of the glass? b. Your friend claims that the rotational speed of the water would have to increase for the cross section to be modeled b = 0.. Is our friend correct? Eplain our reasoning. (, 3.) (, 3.) (0, 0) Maintaining Mathematical Proficienc Evaluate the epression when n = 3 and =. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons 3. n n n + Chapter Graphing Quadratic Functions