# 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 18E

a.

To determine

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

Expert Solution

the quadratic function is expressed in standard form as  f(x)=3(x1)2+1

### Explanation of Solution

Given: The function is f(x)=3x2+6x2

Calculation:

The quadratic function f(x)=3x2+6x2 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)=(3x2+6x2)f(x)=3(x22x)2                           [factor 3 from the x terms] f(x)=3(x22x+1)2(31)        [complete the square : add 1 inside parentheses, subtract (31) outside] f(x)=3(x1)2+1                                 [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)=3(x1)2+1

b.

To determine

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

Expert Solution

the vertex is (h,k)=(1,1) .

The x-intercept is 3±3

y-intercept=f(0)=2

### Explanation of Solution

Given: The function is f(x)=3x2+6x2

Calculation:

The quadratic function f(x)=3x2+6x2 is expressed in standard form as:

f(x)=3(x1)2+1

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(1,1) .

The y-intercept is given when x=0 , so y-intercept=f(0)=2

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (h,k)=(1,1) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=2 . To find the x-intercepts , set f(x)=0 and factor the resulting equation.

x=b±b24ac2aHere, a=3,b=6,c=2x=6±(6)24(3)22(3)x=6±36246x=6±126=6±236=3±3

The x-intercept is 3±3

c.

To determine

Expert Solution

### Explanation of Solution

Given: The function is f(x)=3x2+6x2

Graph:

The standard form of the function is:

f(x)=3(x1)2+1

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (1,1) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=2 and the x-intercept is 3±3 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=3x2+6x2

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