Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Question

Chapter 3.1, Problem 8E

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+6x−1)

Calculation:

The quadratic function f(x)=(3x2+6x−1) is expressed in standard form as:

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

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

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

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+6x−1)

Calculation:The standard form of the function is:

f(x)=3(x+1)2−4

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

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+6x−1)

Calculation:

The standard form of the function is:

f(x)=3(x+1)2−4

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

Chapter 3.1, Problem 7E

Chapter 3.1, Problem 9E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 3.1 - To put the quadratic function f(x)=ax2+bx+c in...Ch. 3.1 - The quadratic function f(x) = a(x - h)2 + k is in...Ch. 3.1 - The graph of f(x) = 3(x - 2)2 - 6 is a parabola...Ch. 3.1 - The graph of f(x) = -3(x - 2)2 - 6 is a parabola...Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

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