(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and hor

(a) state the domain of the function, (b) identify all...

Key Concepts

Domain of Rational Functions

The domain of a rational function consists of all real numbers except where the denominator is equal to zero. Determining the domain typically involves setting the denominator equal to zero and excluding those solutions, ensuring the function remains defined for all allowed values.

Intercepts of the Function

Intercepts, including both x-intercepts and y-intercepts, indicate where the graph of the function crosses the axes. Finding the y-intercept involves substituting x = 0, while finding the x-intercepts requires setting the numerator equal to zero and ensuring these values are within the domain.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of the function is zero and the function approaches infinity or negative infinity. They represent the boundaries where the function is undefined and are identified by solving the equation of the denominator equal to zero.

Horizontal Asymptotes

Horizontal asymptotes describe the end-behavior of a rational function as x approaches positive or negative infinity. They are found by comparing the degrees of the numerator and denominator; when these degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, while differing degrees lead to different asymptotic behaviors.

Graph Sketching Techniques

Graph sketching for rational functions involves combining the analysis of the domain, intercepts, and asymptotic behavior with plotting additional points to accurately depict the function's curve. This comprehensive approach ensures that key features such as turning points, relative extremes, and behavior near asymptotes are clearly illustrated.

Transcript

00:08 Again, as graphing rational functions, we want to find some of the key features, and we want to start out by factoring the numerator and the denominator that will tell us several of the key features.

00:20 So if we factor this numerator, we're looking for two numbers that multiply to give us negative 12, that add to give us negative 8.

00:37 So that does not factor.

00:50 I can, let's look at the denominator, two numbers that multiply to give me negative 4 that add to give me negative 3.

01:00 It's going to be negative 4 and positive 1.

01:07 So i'm going to take the 2x squared.

01:10 I'm going to rewrite this negative 3x using negative 4 and positive 1, and then just bring down my negative 2.

01:19 So all i've done is i've rewritten the negative 3x using these two numbers that i came up with here.

01:24 And then i'm going to factor out a common factor by grouping.

01:31 I'm going to take out a two, or factor out of 2x.

01:35 That leaves me with x minus 2 and a 1, because they don't have anything else in common.

01:42 So i know my denominator factors to be 2x plus 1 times x minus 2.

01:54 My numerator don't think factors.

01:59 So let's go ahead and domain restrictions are going to be at 2 and at negative 1 half.

02:14 So our domain is the set of all real numbers not equal to 2 or negative 1 half.

02:19 I think we're going to have a vertical asymptote at x equals 2 and a vertical asymptote at x equals negative 1 half.

02:36 It does factor negative 6 and negative 2.

02:41 Had a brain hiccup there.

02:43 So 3x squared minus 6x plus minus 2x plus 4.

02:51 And we're going to factor by grouping.

02:54 3x times x minus 2 minus 2 times x minus 2...

Please give Ace some feedback