Rational Equations

A rational equation is an equation that can be written as

 Where N(x) and D(x) are polynomials.

 

 The Domain of a rational function includes all real numbers except for any values where D(x)=0.  this is because if D(x)=0, The function would be undefined.  However, the zeros of D(x) are still important to rational functions.  At the zeros of D(x), there is an imaginary line known as a vertical asymptote.  As the Graph approaches the asymptote, the Y values will go to infinity and negative infinity but never cross the line where D(x)=0.  

Graphs can also have horizontal asymptotes.   

If the degree of N(x) is Greater than the degree of D(x), then there is no horizontal asymptote.

If the degree of N(x) is less than the degree of D(x), then there is a horizontal asymptote at Y=0.

If the degree of N(x) is equal to the degree of D(x),then the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator.

Other points of interest include:

X-intercepts: when f(x)=0 or more simply, when N(x)=0.

Y-intercept: f(0)

Example:

Based on the equation, we can determine that there will be a vertical asymptote at x=3 and a horizontal asymptote at y=2.
By solving for f(0), we can determine that the y-intercept will be at (0, 0)
By solving for f(x)=0, we can determine that there is an x-intercept at (0, 0)

In order to describe what the graph is doing at the vertical asymptote, we must use limit notation.

Saying that the graph decreases as it moves toward the Asymptote from the left, we say

The Superscript - on the 3 indicates the left side of the graph.
Thus in order to describe the Right side of the graph we use a superscript + on the 3.

Difference Quotients

A Difference Quotient is the formula used to find the slope of the secant line of two points on a graph, or the tangent line of a single point on a graph.

Simply, for any function f(x) the difference quotient for that function is

With h being any nonzero value in the domain of the original function.

How does one form this formula?

First take a simple, curvy graph and pick any point, lets call this point (a, f(a)) with "a" being the x value of the point, and "f(a)" being the output, or y value.
Then add any value "h" to "a" and plot a point on the graph there, creating the point (a+h, f(a+h)). Your graph will look something like this.

Since you now have 2 points, and therefore a line secant to this graph, let us find this lines slope by using the formula of the slope of a line.

And there it is, proven right above, in simplest terms the difference quotient is merely the formula of the slope of a line. But you can do so much more with the difference quotient than a plain slope formula, you can manipulate it, as long as you have a knowledge of limits, to find the slope of the tangent line of any point on a graph. Although that is for next year in calculus, for now you only need the basic knowledge of how it is formed and how it could possibly be applied.

Also here are some good tips.

• If you get 1, you probably did something very wrong.
• Stuff is going to cancel each other out in the numerator, so if it doesn't happen you distributed wrong in the f(x+h)
• It's going to divide so that there is no denominator, if it doesn't you distributed wrong again
• And as always, if it just looks funky double check your work!

By Jacob D on Nick C's account because mine was being weird.