Chapter 4.6, Problem 19E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Analyzing the Graph of a Function In Exercises 5-34, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = x 2 − 6 x + 12 x − 4

To determine

To graph: The function y=x26x+12x4.

Explanation

Given: The function y=x2âˆ’6x+12xâˆ’4.

Graph:

Consider the provided function.

The function being a polynomial has a domain as all the real numbers except for 4.

The provide function can be rewritten as:

y=(xâˆ’2)+4xâˆ’4

The function will return all possible values less than âˆ’2 and all values above 6. This implies that the range of the function is (âˆ’âˆž,âˆ’2]âˆª[6,âˆž).

Now find the x and y intercepts by equating f(x) and x to zero respectively to obtain:

There y-intercept is (0,âˆ’3) and there is no x-intercept.

The function has a vertical asymptotes x=4 as the denominator is not defined at those points.

The function can be rewritten as:

y=(xâˆ’2)+4xâˆ’4

This implies that the line y=xâˆ’2 is the slant asymptote of the provided function.

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

x2âˆ’8x+12(xâˆ’4)2=0x2âˆ’8x+12=0x=2,6

This gives four test intervals (âˆ’âˆž,2),(2,4),(4,6),(6,âˆž).

Let 0âˆˆ(âˆ’âˆž,2).

f'(0)=(0)2âˆ’8(0)+12((0)âˆ’4)2>0

The function is increasing in this interval.

Let 3âˆˆ(2,4).

f'(3)=(3)2âˆ’8(3)+12((3)âˆ’4)2<0

The function is decreasing in this interval.

Let 5âˆˆ(4,6).

f'(5)=(5)2âˆ’8(5)+12((5)âˆ’4)2<0

The function is decreasing in this interval

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started