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Chapter 4.6, Problem 16E
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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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Chapter
Section
BuyFindarrow_forward

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 = 4 x 2 + 1

To determine

To graph: The function y=4x2+1.

Explanation

Given: The function y=4x2+1.

Graph:

Consider the provided function.

The function being a rational function has a domain as all the real numbers except 0.

The function will only return values that are greater than 1. This implies that the range of the function is (1,âˆž).

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

There are no y-intercepts or x-intercepts.

The function has one vertical asymptote x=0 as the denominator is not defined at those points.

Also,

limxâ†’âˆž(4x2+1)=4limxâ†’âˆž1x2+1=4(0)+1=1

Hence, the horizontal asymptote is y=1.

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

âˆ’8x3=0

This can never be zero and thus the function has no critical points.

This gives two test intervals (âˆ’âˆž,0),(0,âˆž).

Let âˆ’1âˆˆ(âˆ’âˆž,0).

f'(âˆ’1)=âˆ’8(âˆ’1)3=8>0

The function is increasing in this interval

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