Graph the rational functions in Exercises 103-108 . Include the graphs and equations of the asym

step 1 Okay, so here we have the function x squared minus 1 over 2x plus 4. And first of all, we just w

Graph the rational functions in Exercises 103-108 . Include...

Key Concepts

Rational Functions

Rational functions are expressions that represent the quotient of two polynomials. They provide insights into many behaviors of functions, such as discontinuities and asymptotic behavior, and are central to algebra and calculus studies. Understanding their structure is essential for analyzing limits and end behavior.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at values that make the denominator equal to zero, provided these values do not also cancel with a factor in the numerator. These asymptotes indicate points where the function’s value becomes unbounded and provide critical information about the function’s discontinuities.

Oblique (Slant) Asymptotes

Oblique asymptotes, also known as slant asymptotes, arise when the degree of the numerator is exactly one greater than that of the denominator. They are determined by performing polynomial division, where the quotient represents the equation of the asymptote. This concept is key to understanding the end behavior of rational functions.

Graphing Rational Functions

Graphing rational functions involves plotting key features such as intercepts, asymptotes, and points of discontinuity. By analyzing these components, including the behavior near asymptotes and at infinity, one can accurately sketch the function’s graph. This process integrates multiple concepts in algebra and calculus to build a comprehensive picture of the function’s behavior.

Transcript

00:01 Okay, so here we have the function x squared minus 1 over 2x plus 4.

00:13 And first of all, we just want to see what type of asymptotes we have.

00:18 Since the degree of the numerator is one more than the denominator, we're going to have a slant asymptote.

00:22 So we'll use synthetic division here.

00:30 And so actually, one thing we can do to make this a little easier is i can write this as one -half, because the one half isn't going to make a difference.

00:44 So i'm just factoring out of two from the denominator.

00:48 It will just make the synthetic division a little bit easier having a monic polynomial here.

00:54 What i mean is the leading coefficient of each polynomial to be one.

01:04 So let's divide x plus 2 into x squared minus 1.

01:11 So x goes into x squared x times.

01:14 So we have x plus 2x.

01:21 So we're going to have, we're going to be subtracting these.

01:25 That's 0.

01:26 We have minus 2x, negative 1 minus 0, it's minus 1.

01:31 And then x goes into minus 2x minus 1, negative 2 times.

01:35 That's minus 2.

01:36 So we'll have negative 2 x minus 4.