(a) Explanation Definitions used: Properties of the graph of a quadratic function: The graph of the quadratic function f ( x ) = a x 2 + b x + c ; a ≠ 0 is a parabola then, 1. Vertex = ( − b 2 a , f ( − b 2 a ) ) . 2. Axis of symmetry: the vertical line x = − b 2 a 3. Parabola opens up if a > 0 , the vertex is a minimum point. 4. Parabola opens down if a < 0 , the vertex is a maximum point. The x-intercepts of a quadratic function: 1. If the discriminant b 2 − 4 a c > 0 , the graph of f ( x ) = a x 2 + b x + c has two distinct x -intercepts, so it crosses the x -axis in two places. 2. If the discriminant b 2 − 4 a c = 0 , the graph of f ( x ) = a x 2 + b x + c has one x -intercept, so it touches the x -axis at its vertex. 3. If the discriminant b 2 − 4 a c < 0 , the graph of f ( x ) = a x 2 + b x + c has no x -intercepts, so it does not cross or touch the x -axis. Calculation: The given function is f ( x ) = 9 2 x 2 + 3 x + 1 . Compare the function f ( x ) = 9 2 x 2 + 3 x + 1 with the standard form f ( x ) = a x 2 + b x + c . Here, a = 9 2 , b = 3 , c = 1 . As a = 9 2 > 0 , the graph opens up. The x -coordinate of the vertex is x = − b 2 a . Evaluate the x -coordinate by substituting a = 9 2 , b = 3 in x = − b 2 a as follows. x = − 3 2 ( 9 2 ) = − 3 × 2 18 = − 1 3 Thus, the minimum value occurs at x = − 1 3 (b) To determine The domain and range of f ( x ) = 9 2 x 2 + 3 x + 1 . (c) To determine The intervals where the quadratic function f ( x ) = 9 2 x 2 + 3 x + 1 is increasing and decreasing.

Pearson eText Algebra and Trigonometry -- Instant Access (Pearson+)

11th Edition

ISBN: 9780136872689

Author: Michael Sullivan

Publisher: PEARSON+

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Question

Chapter 4, Problem 12RE

(a)

To determine

To sketch: The graph of the function f(x)=92x2+3x+1 by finding whether the graph opens up or down, and by finding the vertex, axis of symmetry, y-intercept and x-intercepts.

(b)

To determine

The domain and range of f(x)=92x2+3x+1.

(c)

To determine

The intervals where the quadratic function f(x)=92x2+3x+1 is increasing and decreasing.

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Chapter 4, Problem 11RE

Chapter 4, Problem 13RE

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Chapter 4 Solutions

Pearson eText Algebra and Trigonometry -- Instant Access (Pearson+)

Ch. 4.1 - Prob. 1AYU Ch. 4.1 - Prob. 2AYU Ch. 4.1 - Prob. 3AYU Ch. 4.1 - Prob. 4AYU Ch. 4.1 - Prob. 5AYU Ch. 4.1 - Prob. 6AYU Ch. 4.1 - Prob. 7AYU Ch. 4.1 - Prob. 8AYU Ch. 4.1 - Prob. 9AYU Ch. 4.1 - Prob. 10AYU

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The graph of the function f ( x ) = 9 2 x 2 + 3 x + 1 by finding whether the graph opens up or down, and by finding the vertex, axis of symmetry, y -intercept and x -intercepts.

To sketch: The graph of the function f(x)=92x2+3x+1 by finding whether the graph opens up or down, and by finding the vertex, axis of symmetry, y-intercept and x-intercepts.

The domain and range of f(x)=92x2+3x+1.

The intervals where the quadratic function f(x)=92x2+3x+1 is increasing and decreasing.

Want to see the full answer?

Chapter 4 Solutions