(a) Explanation Graph of the quadratic function: 1. “The graph of the function f ( x ) = a x 2 + b x + c is a parabola that opens upward, and the vertex is minimum, if a > 0 and the parabola opens downward, and the vertex is minimum, if a < 0 . 2. The vertex is at the point ( ( − b 2 a ) , f ( − b 2 a ) ) ; 3. The axis of symmetry of the parabola is x = − b 2 a .” Calculation: The given function y = x 2 + 8 x + 19 is in the form of a x 2 + b x + c , where a = 1 , b = 8 and c = 19 . As the value of a is greater than zero, that is 1 > 0 or positive, the graph of the function is a parabola that opens upward and has a minimum point. To find the y -intercept, set x = 0 and solve for y . Substitute x = 0 in y = x 2 + 8 x + 19 , y = x 2 + 8 x + 19 = ( 0 ) 2 + 8 ( 0 ) + 19 = 19 Therefore, the y intercept is ( 0 , 19 ) . It is known that the x coordinate of the vertex is x = − b 2 a and the vertex of the parabola is ( ( − b 2 a ) , f ( − b 2 a ) ) . The x coordinate of the vertex can be calculated as follows, x = − b 2 a = − 8 2 ( 1 ) = − 4 Substitute x = − 4 in the function y = x 2 + 8 x + 19 and obtain the value of f ( ( − b 2 a ) ) (b) To determine To find: Whether the graph on the window [ − 10 , 10 ] by [ − 10 , 10 ] gives the complete graph or not.

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)

5th Edition

ISBN: 9780134179025

Author: Ronald J. Harshbarger, Lisa S. Yocco

Publisher: PEARSON

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Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 3.1, Problem 11E

(a)

To determine

To sketch: The graph of the quadratic function y=x2+8x+19.

(b)

To determine

To find: Whether the graph on the window [−10,10] by [−10,10] gives the complete graph or not.

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Chapter 3.1, Problem 10E

Chapter 3.1, Problem 12E

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College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)

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The graph of the quadratic function y = x 2 + 8 x + 19 .

Solution Summary: The author illustrates the graph of the quadratic function y =x2+8x+19.