   Chapter 3.R, Problem 22E

Chapter
Section
Textbook Problem

# 17-28 Use the guidelines of Section 3.5 to sketch the curve. y = 1 x 2 − 1 ( x − 2 ) 2

To determine

To sketch:

The graph of the function y=1x2-1x-22

Explanation

1) Concept:

i) A domain is the set of xvalues that satisfy the function.

ii) To find x-intercept, put y=0,and to find y-intercept, put x=0in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if  f-x=fx, then it is an even function, so it has y-axis symmetry. If  f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if  f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0 and decreasing if f'x<0 in that particular interval.

vi) The number f(c)is a local maximum value of f if fcf(x)when x is near c and is a local minimum value of fif fc f(x)when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

The function y=1x2-1x-22

3) Calculation:

A) Domain

The given function is y=1x2-1x-22= x-22-x2x2x-22

The domain of this function is set for all values of x except for those where the denominator becomes zero.

The zeros of the denominator is x=0 and x=2

Therefore, the domain of the function is (-, 0)(0 ,2)(2 ,)

B) Intercepts

For y intercept plug x=0  in the given function and solve it

As the function is not defined for x=0, there is no y-intercept

Now for x-intercept set y = 0 then equation becomes

1x2-1x-22=0

We get,

1x2=1x-22

If and only if,

x2=x-22

x2=x2-4x+4

-4x+4=0

x=1

Therefore, x-intercept is x=1

C) Symmetry

For symmetry, replace x by (-x)

f(-x)=1-x2-1-x-22

Since, f-x-fxand fxf(-x)

That means there is no symmetry.

Therefore f is neither an even nor an odd function

D) Asymptote

limx±1x2-1x-22=0

So y=0 is horizontal asymptote

limx0- 1x2-1x-22= and limx0+1x2-1x-22=

Therefore, x=0  is vertical asymptote

### 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

#### Evaluate the definite integral. 02(x1)e(x1)2dx

Single Variable Calculus: Early Transcendentals, Volume I

#### Expand each expression in Exercises 122. (x32x2+4)(3x2x+2)

Finite Mathematics and Applied Calculus (MindTap Course List)

#### Simplify: 7296

Elementary Technical Mathematics

#### True or False: The domain is all numbers x for which f(x) is defined.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 