# 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 21E

a.

To determine

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

Expert Solution

the quadratic function is expressed in standard form as  f(x)=4(x+2)2+19

### Explanation of Solution

Given: The function is f(x)=(4x216x+3)

Calculation:

The quadratic function f(x)=(4x216x+3) 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)=(4x216x+3) f(x)=4(x2+4x)+3                            [factor 4 the x terms] f(x)=4(x2+4x+4)+3(44)       [complete the square: add 4 to parentheses, subtract (44) outside] f(x)=4(x+2)2+19                             [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)=4(x+2)2+19

b.

To determine

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

Expert Solution

the vertex is (h,k)=(2,19) .

The x-intercept is 2±192

y-intercept=f(0)=3

### Explanation of Solution

Given: The function is f(x)=(4x216x+3)

Calculation:

The quadratic function f(x)=(4x216x+3) is expressed in standard form as:

f(x)=4(x+2)2+19

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(2,19) .

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (h,k)=(2,19) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=3 . To find the x-intercepts , set f(x)=0 and factor the resulting equation.

x=b±b24ac2aHere, a=4,b=16,c=3x=16±(16)24(4)32(4)x=16±256+488x=16±3048=16±4198=4±192=2±192

The x-intercept is 2±192

c.

To determine

Expert Solution

### Explanation of Solution

Given: The function is f(x)=(4x216x+3)

Graph:

The standard form of the function is:

f(x)=4(x+2)2+19

From the standard form it is observed that the graph is a parabola that opens upward and has vertex (2,19) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=3 and The x-intercept is 2±192 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(4x216x+3)

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