# To Express: The quadratic function in standard form.

### 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 20E

a.

To determine

## To Express: The quadratic function in standard form.

Expert Solution

the quadratic function is expressed in standard form as  f(x)=1(x+14)24916

### Explanation of Solution

Given: The function is f(x)=(2x2+x6)

Calculation:

The quadratic function f(x)=(2x2+x6) 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 upwards if a>0 .

Solve the function:

f(x)=(2x2+x6)f(x)=(22x2+12x62)=(x2+12x3)   [divide the function by 2] f(x)=1(x2+12x)3                               [factor 1 the x terms] f(x)=1(x2+12x+116)3(1116)       [complete the square: add 116 to parentheses, subtract (1116) outside] f(x)=1(x+14)24916                              [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)=1(x+14)24916

b.

To determine

### To Find: The vertex, x−intercept and y−intercept

Expert Solution

the vertex is (h,k)=(14,4916) .

x-intercept is 2 and 32

y-intercept=f(0)=6

### Explanation of Solution

Given: The function is f(x)=(2x2+x6)

Calculation:

The quadratic function f(x)=(2x2+x6) is expressed in standard form as:

f(x)=1(x+14)24916

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(14,4916) .

The y-intercept is given when x=0 , so x-intercept is 2 and 32

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (h,k)=(14,4916) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=6 . To find the x-intercepts , set f(x)=0 and factor the resulting equation. It is observed that the x-intercept is 2 and 32 .

c.

To determine

Expert Solution

### Explanation of Solution

Given: The function is f(x)=(2x2+x6)

Graph:

The standard form of the function is:

f(x)=1(x+14)24916

From the standard form it is observed that the graph is a parabola that opens upward and has vertex (14,4916) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=6 and there is no x-intercept . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(2x2+x6)

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