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

step 1 Let's create a graph of the function f of x equals 1 over x plus 2. To start, we'll need to plug

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

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Key Concepts

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. They are characterized by their domains, which are determined by the values that make the denominator zero, and by their asymptotic behavior, which gives insights into how the function behaves near points of discontinuity and at the extremes of the domain.

Vertical Asymptotes

Vertical asymptotes are vertical lines that the graph of a function approaches but never touches or crosses, typically occurring at values of x that make the denominator zero while leaving the numerator nonzero. They indicate points of undefined value in the function and are crucial for understanding the graph's behavior near discontinuities.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values become extremely large or small. They represent the value that the function approaches at the ends of the graph, and are typically determined by comparing the degrees of the numerator and denominator in a rational function.

Transcript

00:01 Let's create a graph of the function f of x equals 1 over x plus 2.

00:04 To start, we'll need to plug in some different points and find where they lie on the graph.

00:08 So i'm going to start by plotting f of negative 2.

00:12 This will be 1 over negative 2 plus 2 and negative 2 plus 2 is 0.

00:18 So we have division by 0, which is strictly not allowed.

00:21 It's undefined.

00:23 Therefore we know there can't be a point here at f of negative 2.

00:27 So i'm going to put a line there.

00:29 That is a line which no part of the graph is allowed to touch because that would be undefined.

00:35 Next up, let's try going on either side of this line, such as f of 1, or negative 1 rather.

00:43 Well, that will be 1 over negative 1 plus 2 or 1 over 1, which is just 1, allowing us to put a point here.

00:52 Then if we try f of 0, that's going to give us 1 half, putting a point here, then, well, suppose we want to get a bit closer to our vertical line here that is a bit closer to negative 2, but we don't want to quite touch it.

01:10 Well, let's try f of negative 1 and 1 half.

01:16 So that will give us, well, 1 over negative 1 .5 plus 2 is 1 over 1⁄2, which is 2, putting a point here.

01:27 We've got enough information to draw this out...

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