# The coordinates of the vertex of a quadratic function f. ### 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 8E

a.

To determine

## To Find: The coordinates of the vertex of a quadratic function f.

Expert Solution

The coordinates of the vertex are (1,4)

### Explanation of Solution

Given: The function is f(x)=(3x2+6x1)

Calculation:

The quadratic function f(x)=(3x2+6x1) is expressed in standard form as:

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

The parabola opens upwards if a>0 .

Solve the function:

f(x)=(3x2+6x1)f(x)=3(x2+2x)1                         [factor 3 from the x terms] f(x)=3(x2+2x+1)131           [complete the square : add 1 inside parentheses, subtract 31 outside]  f( x )=3 (x+1) 2 4                            [factor and multiply]

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

Therefore, the coordinates of the vertex are (h,k)=(1,4)

b.

To determine

### To Find: The maximum or minimum value of f.

Expert Solution

The minimum value is f(1)=4

### Explanation of Solution

Given:The function is f(x)=(3x2+6x1)

Calculation:The standard form of the function is:

f(x)=3(x+1)24

If a>0 , then the minimum value of f is f(h)=k , and is represented as f(1)=4

Since, the coefficient of x2 is positive, f has a minimum value. The minimum value is f(1)=4

c.

To determine

### To Find: The domain and range of f.

Expert Solution

the domain is from (,) and the range is [4,)

### Explanation of Solution

Given: The function is f(x)=(3x2+6x1)

Calculation:

The standard form of the function is:

f(x)=3(x+1)24

It is observed from the standard form of the function thatthe domain is from (,) and the range is [4,)

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!