Graph each rational function. Give the equations of the vertical and horizontal asymptotes. See

step 1 Let's make a graph of this function f of x equals 1 over x minus 2. To begin, we'll have to find

Graph each rational function. Give the equations of the...

Key Concepts

Rational Functions

A rational function is a function represented by the ratio of two polynomials. Its behavior is determined by the degrees of the numerator and denominator, which influence aspects such as asymptotic behavior and discontinuities.

Vertical Asymptote

A vertical asymptote occurs at a value of x that makes the denominator zero (provided the numerator isn’t also zero at that point), causing the function to approach infinity or negative infinity. It represents a line that the graph approaches but never touches or crosses.

Horizontal Asymptote

A horizontal asymptote describes the behavior of a rational function as x tends to positive or negative infinity. It is determined by comparing the degrees of the numerator and denominator; if the degree of the numerator is less than that of the denominator, the asymptote is y = 0, indicating the function's end behavior.

Transcript

00:01 Let's make a graph of this function f of x equals 1 over x minus 2.

00:04 To begin, we'll have to find some ordered pairs, that is, find some points that we can plot.

00:09 So we'll plug in some different values of x.

00:12 Starting with x equals 0.

00:14 So f of 0 is going to give us 1 over, well, 0 minus 2, so just 1 over negative 2, negative 1 1⁄2.

00:23 So we can put a point here at 0, negative 1⁄2.

00:27 All right, now how about f of 1? well, that will give us 1 over 1 minus 2 or 1 over negative 1.

00:35 So just negative 1, giving us the point 1 negative 1.

00:39 Then, f of 2, well, that would be a problem because then we'd have 1 over 2 minus 2, and 2 minus 2 is 0.

00:47 We can't divide by 0.

00:49 So we're not going to have any points at x equals 2.

00:52 I'm going to draw a large vertical line there to show that we can't plot points there.

00:57 But we could put points at other places, such as f of negative 1.

01:03 That will give us 1 over negative 1 minus 2, or, well, negative 1 3rd.

01:09 So we'll have something like this, and it's pretty clear that the graph is taking shape to look like that.

01:16 Now, what if we went on the other side of the big green line, that is on the other side of x equals 2? like, for example, x equals 3.

01:25 Well, that would be f of 3, which is 1 over 3 minus 2, or just 1 over 1...

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