Drawing a Curvillinear Circle With Asymptote

Functions, Graphs, and Limits

Intro

Topics
Limits
Functions Are Your Friend
Graphing and Visualizing Limits
Piecewise Functions and Limits
One-Sided Limits
Limits via Tables
Limits via Algebra
All About Asymptotes
Vertical Asymptotes
Finding Vertical Asymptotes
Vertical Asymptotes vs. Holes
Limits at Infinity
Natural Numbers
Limits Approaching Cypher
Estimating a Circumvolve
The Cantor Set and Fractals
Limits of Functions at Infinity
Horizontal, Slant, and Curvilinear Asymptotes
Horizontal Asymptoes
Slant Asymptotes
Curvilinear Asymptotes
Finding Horizontal/ Slant/ Curvilinear Asymptotes
How to Draw Rational Functions from Scratch
Comparing Functions
Power Functions vs. Polynomials
Polynomials vs. Logarithmic Functions
Manipulating Limits
The Basic Backdrop
Multiplication by a Constant
Adding and Subtracting Limits
Multiplying and Dividing Limits
Powers and Roots of Limits
In the Real World
I Like Abstruse Stuff; Why Should I Intendance?
How to Solve a Math Trouble

Examples
Exercises
Quizzes
Terms
Handouts
Best of the Spider web
Table of Contents

At a Glance - How to Depict Rational Functions from Scratch

We now have plenty tools to draw some complicated functions from scratch. Now we know how graphing calculators exercise information technology, and why they require the energy of four triple-A'south.

When drawing a rational function f(x) from scratch, we demand to know a lot of information, which can be nicely grouped into iii big chunks.

Nosotros need to know where f has vertical asymptotes and/or holes.
We need to know the horizontal/slant/curvilinear asymptotes of f, if whatever.
We need to know about values of f. We found where f is undefined when we found the vertical asymptotes and holes; at present we demand to know where f(ten) is 0, positive, and negative. We also want to know f(0), also called the y-intercept.

Examples
Exercises

Case 1

Graph the function

Go through the categories.

Vertical asymptotes / holes: f has no vertical asymptotes. Information technology has a hole at (2,1).
Horizontal/etc. asymptotes: f doesn't accept any of these, since it's 2 everywhere, as opposed to "approaching'' 2.
The domain of f: f is undefined at x = ii, and one everywhere else..

Taking this information, nosotros draw the graph:

Example 2

Graph the function .

f has a vertical asymptote at x = 0, and no holes. We can start cartoon:

Since the degree of the numerator and denominator are the aforementioned, f has a horizontal asymptote at y = three:

f is 0 when its numerator is 0 and its denominator is not. This occurs when.

The simply thing left is to find where f is negative and positive. When the numerator and denominator are both negative, therefore f is positive. When , the numerator is positive and the denominator is negative, so f is negative.

We now have enough information to go a rough sketch of the piece of f that lies to the left of the vertical asymptote at y = 0.

When x > 0, the numerator and denominator are both positive and f is positive. Since f must approach its asymptotes, f looks like this:

We didn't demand to worry most f(0) for this function, since f is undefined at 0.

A number line is a useful tool for figuring out where a office is negative and positive, and we'll use this tool in i of the examples.

Wherever f is 0 or undefined, its sign has the ability to change. Draw a number line and marker all the values of ten where f is 0 or undefined. From this, observe the sign of f in between those marked values.

Example iii

Graph the role

Start we need to factor.

There is no way to simplify this function; no terms volition cancel. Nosotros find no holes, but vertical asymptotes at 10 = -3 and x = -1:

Since the caste of the numerator and degree of the denominator are the same, we have a horizontal asymptote at:

The function is 0 when the numerator is 0, which occurs at x = -2 and x = 1:

Now nosotros demand to figure out where the function is negative and where it is positive, which we can do with a number line:

Finding f(0):

Nosotros can add together this point to the graph:

We know some points of f, nosotros know where information technology is 0, negative, and positive, and nosotros know information technology must approach its asymptotes, and so f must look like this:

We haven't explained how we know that f has the shape it has, and we won't get to that until we starting time talking about concavity. For at present, think near drawing the functions to make them "polish," similar MJ's criminal.

Remember that intercepts are where a function crosses the axes. The y-intercept is f(0), if that exists. The x-intercepts are all values of x where the office is 0.

Example 4

Graph the office

Label all asymptotes, intercepts, and holes.

Kickoff we cistron:

We take a vertical asymptote at x = -1 and a hole at (1,eight):

We have no horizontal asymptote, merely since the degree of the numerator is one greater than the degree of the denominator we do have a slant asymptote, which nosotros can find via long segmentation: