BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

15th Edition

ISBN: 9781608408405

Author: HOUGHTON MIFFLIN HARCOURT

Publisher: Cengage Learning

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Question

Chapter 4.7, Problem 36E

To determine

To find: the maximum and minimum value of the given function. Also describe the domain and range of the function and the interval where the graph is increasing and decreasing.

Expert Solution & Answer

Answer to Problem 36E

The function is increasing on $(3,∞)$ and decreasing on $(−∞,3)$ .

The function has minimum value and the minimum value is $f(x)=−147$ .

Domain of the function is set of all real numbers.

Range of the function is $f(x)≥−147$ .

Explanation of Solution

Given information:

Given function is

$f(x)=3(x−10)(x+4)$

Concept used:

The function $g(x)=a(x+h)2+k$ has minimum value if $a>0$ and the parabola opens up.

The function $g(x)=a(x+h)2+k$ has maximum value if $a<0$ and the parabola opens down.

The minimum or maximum value occurs at $x=h$ and the minimum or maximum value is $g(x)=k$ .

Where $(h,k)$ is the vertex of parabola.

Domain of a function $y=f(x)$ is all $x$ values for which $y=f(x)$ is defined.

Range of a function $y=f(x)$ is all $y$ values for which $y=f(x)$ is defined.

Calculation:

Consider the given function.

$f(x)=3(x−10)(x+4)$

The above function can be written as

$f(x)=3(x2+4x−10x−40)=3x2−18x−120=3(x2−6x+9)−147=3(x−3)2−147$

The function $f(x)=3(x−3)2−147$ is of the form $g(x)=a(x+h)2+k$ where $a=3,h=−3,k=−147$ .

So, here $a>0$ , the parabola opens up.

The function has minimum value at $x=h=−3$ .

Minimum value of the function is $k=−147$ .

Domain of the function is set of all real numbers.

Range of the function is $f(x)≥−147$ .

Now, sketch the graph of the function.

BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015, Chapter 4.7, Problem 36E

The function is increasing to the right of $x=3$ that is on $(3,∞)$ .

The function is decreasing to the left of $x=3$ that is on $(−∞,3)$ .

Chapter 4.7, Problem 35E

Chapter 4.7, Problem 37E

Chapter 4 Solutions

BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

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