DISCUSS: Constructing a Rational Function from Its...

DISCUSS: Constructing a Rational Function from Its Asymptotes Give an example of a rational function that has vertical asymptote $x=3 .$ Now give an example of one that has vertical asymptote $x=3$ and horizontal asymptote $y=2 .$ Now give an example of a rational function with vertical asymptotes $x=1$ and $x=-1,$ horizontal asymptote $y=0,$ and $x$ -intercept 4.

Key Concepts

Intercepts

Intercepts, including x-intercepts and y-intercepts, are the points where the graph of the rational function crosses the coordinate axes. For rational functions, x-intercepts are found by setting the numerator equal to zero while ensuring the function remains defined, providing key information for graphing.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of rational functions as x approaches infinity or negative infinity. They are determined by comparing the degrees of the numerator and denominator and help in predicting the end behavior of the function.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another, ensuring the denominator is not zero. They exhibit a range of behaviors including discontinuities, asymptotic behavior, and intercepts which are critical for understanding their overall graph structure.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non-zero, leading the function to diverge to infinity near these points. Analyzing vertical asymptotes is essential for understanding the function's behavior and domain restrictions.

Transcript

00:01 Problem 93 asks us to actually create our own rational expressions from some given criteria.

00:10 So the first situation is they're saying the only thing you need to make sure you have is that the vertical asymptote is x equals three.

00:25 Okay.

00:26 So from our experience, we should know that that has to deal with the denominator and the answer to the question of what makes the bottom.

00:33 Zero needs to be three.

00:36 So the easiest way to express that is just put x minus three.

00:39 As far as the numerator, it could be really anything except for x minus three.

00:46 Because if you put x minus three, it'll cancel with the x minus three on bottom.

00:51 So the simplest thing to put with just to be to make up some number and throw in there.

00:56 So our function could be something like one over x minus three.

01:01 Okay, the second situation, they say again, let's say the vertical asymptote is x equals 3, but now we need to have a horizontal asymptote of y equals 2.

01:18 Okay, so with vertical asymptote, we know that the denominator could be x minus 3.

01:25 With the horizontal asymptote, that has to deal with the degrees of the numerator and denominator comparatively.

01:33 The only way to get a number is when the degrees are the same, so that we need this to be x.

01:43 And to get two, we need the ratio of these two leading coefficients to be two.

01:48 So we could throw it to a top there.

01:51 Now the ratio of these leading coefficients is two over one, which is two.

01:55 Now, you could certainly add a number in here plus some number.

01:59 I'll just put b.

02:01 The only thing you could not do, again, is you would not want this numerary to be factored so that you have x minus three up top, which would cancel.

02:09 So b could not be negative 6, because then that could be factored as 2, x minus 3, the x minus 3 is cancel...

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