Explanation Procedure used: “Graphing a quadratic function of the form f ( x ) = a x 2 + b x + c , a ≠ 0 , use the following steps: Step 1: Determine whether the parabola open up or down. Step 2: Determine the vertex and axis of symmetry. Step 3: Determine the y- intercept, f ( 0 ) = c . Step 4: Determine discriminant, b 2 − 4 a c . 1. If b 2 − 4 a c > 0 , then the parabola has two x- intercept, which are found by solving f ( x ) = 0 ( a x 2 + b x + c = 0 ) 2. If b 2 − 4 a c = 0 , the vertex is the x- intercept. 3. If b 2 − 4 a c < 0 , there are no x- intercept. Step 5: Plot the vertex, y- intercept, and x- intercept(s). Use the axis of symmetry to find an additional point. Draw the graph of the quadratic function”. Definition used: “Any quadratic function f ( x ) = a x 2 + b x + c , a ≠ 0 has a vertex ( − b 2 a , 4 a c − b 2 4 a ) . Because the y- coordinate of the function can be found by evaluating the function at the corresponding x- coordinate, the coordinates of the vertex can be restated as ( − b 2 a , f ( − b 2 a ) ) ”. Formula used: The solutions to the quadratic equation a x 2 + b x + c = 0 , a ≠ 0 , are given by the quadratic formula x = − b ± b 2 − 4 a c 2 a . Calculation: The given function is G ( x ) = − 3 x 2 + 8 x + 2 . By comparing the function G ( x ) = − 3 x 2 + 8 x + 2 with general quadratic equation f ( x ) = a x 2 + b x + c , a ≠ 0 , the value of a = − 3 , b = 8 and c = 2 . Determine whether the parabola is opens up or down. Since the value of a = − 3 < 0 , the parabola opens down. Obtain the vertex and axis of symmetry of the parabola. From the above definition, the vertex of the quadratic function f ( x ) = a x 2 + b x + c , a ≠ 0 is ( − b 2 a , f ( − b 2 a ) ) . Obtain the x- coordinate of the vertex as follows: x = − b 2 a = − ( 8 ) 2 ( − 3 ) = 4 3 That is, the x- coordinate of the vertex is 4 3 . Obtain the y- coordinate of the vertex as follows. y = G ( 4 3 ) = − 3 ( 4 3 ) 2 + 8 ( 4 3 ) + 2 [ ∵ G ( x ) = − 3 x 2 + 8 x + 2 ] = − 16 3 + 32 3 + 2 = 22 3 Therefore, the vertex of the function G ( x ) = − 3 x 2 + 8 x + 2 is ( 4 3 , 22 3 )

3rd Edition

ISBN: 9780321880116

Author: Michael Sullivan III, Katherine R. Struve, Janet Mazzarella

Publisher: PEARSON

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Chapter 10.5, Problem 51E

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To sketch: The graph of the function G(x)=−3x2+8x+2 using its properties and find the domain and range of the quadratic function.

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Chapter 10.5, Problem 50E

Chapter 10.5, Problem 52E

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Chapter 10 Solutions

Elementary & Intermediate Algebra

Ch. 10.1 - If x2 = p, then x = _______ or x = _______.Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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The graph of the function G ( x ) = − 3 x 2 + 8 x + 2 using its properties and find the domain and range of the quadratic function.

Solution Summary: The author explains how to draw the graph of the quadratic function using its properties.

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To sketch: The graph of the function G(x)=−3x2+8x+2 using its properties and find the domain and range of the quadratic function.

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