# 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 12E

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(x5)225

### Explanation of Solution

Given: The function is f(x)=(x2+10x)

Calculation:

The quadratic function f(x)=(x2+10x) 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)=(x2+10x)f(x)=1(x210x)                        [factor 2 from the x terms] f(x)=1(x210x+25)125      [complete the square : add 25 inside parentheses, subtract 125 outside] f(x)=1(x5)225                     [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(x5)225

b.

To determine

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

Expert Solution

the vertex is (h,k)=(5,25) .

x-intercept=0 and 10

y-intercept=f(0)=0

### Explanation of Solution

Given: The function is f(x)=(x2+10x)

Calculation:

The quadratic function f(x)=(x2+10x) is expressed in standard form as:

f(x)=1(x5)225

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(5,25) .

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

The x-intercept is given when y=0 , so x2+10x=0 which implies x(x10)=0

Therefore, x-intercept=0 and 10

c.

To determine

Expert Solution

### Explanation of Solution

Given: The function is f(x)=(x2+10x)

Graph:

The standard form of the function is:

f(x)=1(x5)225

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (5,25) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=0 and the x-intercept=0 and 10 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(x2+10x)

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