BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

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

Answer to Problem 8E

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

Answer to Problem 8E

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

Answer to Problem 8E

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,)

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