# To Express: The quadratic function in standard form.

BuyFind

### Precalculus: Mathematics for Calcu...

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

### Precalculus: Mathematics for Calcu...

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

#### Solutions

Chapter 3.1, Problem 19E

a.

To determine

Expert Solution

## Answer to Problem 19E

the quadratic function is expressed in standard form as  f(x)=2(x5)2+7

### Explanation of Solution

Given: The function is f(x)=(2x220x+57)

Calculation:

The quadratic function f(x)=(2x220x+57) is expressed in standard form as:

f(x)=a(xh)2+k , by completing the square. The graph of the function f is a parabola with vertex (h,k)

The parabola opens downwards if a<0 .

Solve the function:

f(x)=(2x220x+57)f(x)=2(x210x)+57                          [factor 2 from the x terms] f(x)=2(x210x+25)+57(252)    [complete the square : add 25 inside parentheses, subtract (252) outside] f(x)=2(x5)2+7                                [factor and multiply]

On comparing the above equation with standard form f(x)=a(xh)2+k ,

Therefore, the quadratic function is expressed in standard form as  f(x)=2(x5)2+7

b.

To determine

Expert Solution

## Answer to Problem 19E

the vertex is (h,k)=(5,7) .

There is no x-intercept

y-intercept=f(0)=57

### Explanation of Solution

Given: The function is f(x)=(2x220x+57)

Calculation:

The quadratic function f(x)=(2x220x+57) is expressed in standard form as:

f(x)=2(x5)2+7

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(5,7) .

The y-intercept is given when x=0 , so y-intercept=f(0)=57

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (h,k)=(5,7) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=57 . To find the x-intercepts , set f(x)=0 and factor the resulting equation. It is observed that there is no x-intercept

c.

To determine

Expert Solution

### Explanation of Solution

Given: The function is f(x)=(2x220x+57)

Graph:

The standard form of the function is:

f(x)=2(x5)2+7

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (5,7) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=57 and there is no x-intercept . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(2x220x+57)

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