Table of Values A rational function is given. (a) Complete...

Table of Values A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. (c) Determine the horizontal asymptote, based on Tables 3 and 4.
$r(x)=\frac{4 x+1}{x-2}$

Key Concepts

Rational Functions

A rational function is defined as the quotient of two polynomial functions. This concept is central to understanding behaviors such as discontinuities, intercepts, and the occurrence of asymptotes, which are critical in analyzing the overall graph of the function.

Vertical Asymptote

A vertical asymptote occurs at the value(s) of x that make the denominator zero (provided the numerator does not also equal zero at those points). Understanding vertical asymptotes is important because they indicate where a function is undefined and where the function’s values increase or decrease without bound, which significantly affects the graph's shape.

Horizontal Asymptote

A horizontal asymptote represents the value that a rational function approaches as x tends to both positive and negative infinity. This asymptotic behavior is determined by comparing the degrees of the polynomials in the numerator and denominator, providing insight into the end behavior of the function.

Behavior Near Asymptotes

Analyzing the behavior near vertical and horizontal asymptotes involves studying how the function behaves as it approaches these critical values. Using tables of values is a practical method to observe and interpret the tendencies of a rational function, facilitating a deeper understanding of its overall behavior and guiding accurate graph sketching.

Transcript

00:01 Okay, this time they give us the irrational function, r of x, is equal to 4x plus 1 over x minus 2.

00:18 And they want us to fill out several tables to talk about the behavior of our function near its vertical asymptote, which will be given by information from here.

00:30 And our horizontal asymptotes, which we will determine after the tables.

00:38 Okay, so table one for x and r of x, they ask us to put in the input 1 .5 and we see that we get negative 14.

00:55 If you plug that in, 1 .9, it would be negative 86, 1 .99, negative 86, 1 .99, negative 86.

01:04 Okay, it went very much negative very quickly.

01:08 1 .999, a very small change in act.

01:12 And yet now we have negative 8 ,000 -990.

01:17 And what is it getting closer to? it looks like it's getting closer to 2.

01:21 Okay, and that is our vertical asymptote when the denominator will be 0.

01:26 So it's approaching that 0.

01:28 And as we approach it from the left, because we're currently at 1 .5 for the first century, and we could be at 1 .9 of next, 1 .99, 99, 99, and then like 2 right there.

01:50 We are getting very, very negative numbers towards negative infinity.

01:57 What about the second table? does it tell us anything? well, we have x, r of x, and they tell us to plug in 2 .5.

02:08 We end up with 22.

02:10 We put in 2 .1, they give us 94.

02:14 Plug it in and then if you plug in 2 .01 you will get 9004 plug in 2 .001 very close to 2.

02:23 We're approaching larger than it so from the right we started at 2 .5 and we started closing in on 2 from its right side and we end up with 9 ,004 so it seems like we're approaching infinity as we approach our x of 2 from the right we're approaching 2 from there.

02:50 Now, third table.

02:53 Well, actually, we just, we should finish this discussion.

02:56 If we have an asymptote here at negative 2...

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