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 6E

To determine

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

Expert Solution

Answer to Problem 6E

The coordinates of the vertex are (−2,8)

Explanation of Solution

Given: The function is f(x)=−12x2+2x+6

Calculation:

The quadratic function f(x)=−x2+6x−5 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 downward if a<0 .

Solve the function:

f(x)=−12x2+2x+6f(x)=−12(x2−2x)+6 [factor −12 from the x terms] f(x)=−12(x2−2x+4)+6−(−12)⋅4 [complete the square : add +4 inside parentheses, subtract (−12)⋅4 outside] f( x )=− 1 2 (x−2) 2 +8 [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)=(−2,8)

To determine

To Find: The maximum or minimum value of f.

Expert Solution

Answer to Problem 6E

The maximum value is f(−2)=8

Explanation of Solution

Given:The function is f(x)=−12x2+2x+6

Calculation:The standard form of the function is:

f(x)=−12(x−2)2+8

If a<0 , then the maximum value of f is f(h)=k , and is represented as f(−2)=8

Since, the coefficient of x2 is negative, f has a maximum value. The maximum value is f(−2)=8

To determine

To Find: The domain and range of f.

Expert Solution

Answer to Problem 6E

the domain is from (−∞,∞) and the range is (−∞,8]

Explanation of Solution

Given: The function is f(x)=−12x2+2x+6

Calculation:

The standard form of the function is:

f(x)=−12(x−2)2+8

It is observed from the standard form of the function thatthe domain is from (−∞,∞) and the range is (−∞,8]

Chapter 3.1, Problem 5E

Chapter 3.1, Problem 7E

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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