# The statement “The graph of f is a parabola with vertex ( _ _ , _ _ ) ”.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 3.1, Problem 2E

(a)

To determine

## To fill: The statement “The graph of f is a parabola with vertex (__,__) ”.

Expert Solution

The graph of f is a parabola with vertex (h,k) .

### Explanation of Solution

A quadratic function is a polynomial function of degree 2. So a quadratic function is a function of the form f(x)=ax2+bx+c,a0 .

From the definition of standard form of a quadratic equation, a quadratic function f(x)=ax2+bx+c can be expressed in the standard form f(x)=a(xh)2+k by completing the square.

The graph of this function f turns out to be a parabola with vertex set (h,k) .

Hence, the graph of f is a parabola with vertex (h,k) .

(b)

To determine

### To fill: The statement “If a>0 , the graph of f opens ____________. In this case f(h)=k is the ____________ value of f “.

Expert Solution

If a>0 , the graph of f opens upward. In this case f(h)=k is the minimum value of f.

### Explanation of Solution

The standard form of a quadratic function is f(x)=a(xh)2+k .

The graph of this function f turns out to be a parabola with vertex set (h,k) and the parabola opens upward if a>0 .

The maximum or minimum value of f occurs at x=h and if a>0 , then the minimum value of f is f(h)=k .

Hence, if a>0 , the graph of f opens upward. In this case f(h)=k is the minimum value of f.

(c)

To determine

### To fill: The statement “If a<0 , the graph of f opens ____________. In this case f(h)=k is the ____________ value of f “.

Expert Solution

If a<0 , the graph of f opens downward. In this case f(h)=k is the maximum value of f.

### Explanation of Solution

The standard form of a quadratic function is f(x)=a(xh)2+k .

The graph of this function f turns out to be a parabola with vertex set (h,k) and the parabola opens downward if a<0 .

The maximum or minimum value of f occurs at x=h and if a<0 , then the maximum value of f is f(h)=k .

Hence, if a<0 , the graph of f opens downward. In this case f(h)=k is the maximum value of f.

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