Complete the following. (a) Find the domain of f (b) Graph f in an appropriate viewing rectangle

Complete the following. (a) Find the domain of f (b) Graph f...

Key Concepts

Graphing Rational Functions

Graphing rational functions involves integrating several key aspects: accurately determining the domain, identifying vertical and horizontal asymptotes, and investigating the behavior of the function near these asymptotes and in between. A well-drawn graph includes these asymptotes and reflects the function’s behavior over an appropriate viewing rectangle, ensuring a clear visual representation of its characteristics.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of rational functions as x tends towards positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. This concept is important for understanding the long-term behavior of the function.

Domain Analysis

Determining the domain of a function involves identifying all the input values for which the function is defined. For rational functions, this typically means excluding values that make the denominator equal to zero, since division by zero is undefined. This key concept helps in understanding restrictions on the function’s inputs and is fundamental to accurate graphing and analysis.

Rational Functions

Rational functions are quotients of two polynomial functions. They are important because their behavior is determined by the degrees and coefficients of the numerator and denominator, which can result in features such as holes, vertical asymptotes, and horizontal or oblique asymptotes. Studying rational functions involves understanding how changes in the algebraic form affect the graph and properties of the function.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values of x that make the denominator zero, provided these zeros are not canceled by common factors in the numerator. They indicate values where the function grows without bound or approaches negative infinity. Recognizing vertical asymptotes is crucial for understanding the behavior of the function and sketching its graph correctly near points of discontinuity.

Transcript

00:02 For this problem, we are given a rational function, and we're being asked to find some pieces of information about it.

00:11 First being the domain.

00:14 So to find the domain, i always factor the bottom if i can.

00:19 So this is a difference of squares, factors into x minus 3 and x plus 3.

00:26 And then i take that and i see what can make the denominator equal to 0? which would be 3 and negative 3.

00:36 So those are the only two spots where i cannot have x equal to, meaning that my domain is going to be represented as negative infinity to negative 3, union negative 3 to 3, union 3 to positive infinity.

00:57 Then for letter b, they asked us to graph it.

01:00 So i graphed this on my graphing calculator, and i will upload what the slide is going to look like.

01:13 So you can see here, this is what our graph is going to look like.

01:22 For letter c, they asked us to find the horizontal and the vertical asymptotes.

01:28 So vertical asymptotes we find by setting the denominator equal to zero, which we've already done.

01:34 So my vertical asymptotes are three and negative three.

01:43 Horizontal, we find by comparing the degrees of the top and the bottom of this function.

01:49 So if i take a look at this one, i actually have the same thing.

01:54 I have a two on the bottom and a two on the top.

01:59 So because these are equal, we then have the horizontal asymptote, y equals a over b...

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