Rational functions are any functions that can be written as the ratio of two polynomial functions. There are two types of asymptotes for rational functions: vertical asymptotes, which occur at the zeros of the denominator and cannot be crossed, and horizontal asymptotes, which can be crossed. To find the vertical and horizontal asymptotes of a rational function, you examine the degrees of the numerator and denominator polynomials. The domain of a rational function is the set of x-values that make the function defined, while the range is the set of possible y-values produced by the function.

1. Rational FunctionsBy: Katie Thayer

2. A rational function is any function which can be written as the ratio of two polynomial functions.What is a Rational Function?

3. AsymptotesAsymptote - a straight line that is the limiting value of a curveThe red line on the graph indicates that y=x is the asymptote.

4. 2 typesThere are two types of asymptotes. Vertical Asymptote- x=___, ** CAN’T BE CROSSED**. Can be found be the zeros, or x- intercepts, which are in the denominator.Horizontal Asymptote- y=___, ** Can be crossed*. But there are three different ways you can find a horizontal asymptote.

5. VA and HAThe Vertical Asymptote of out function would be, x=-3 and x=1The Horizontal Asymptote is y=0.

6. Three types of Horizontal asymptote.Type 1: when the degree of the denominator- degree being the exponential value of x- is larger than the degree of the numerator, then the HA is y=0.Type 2: When the numerator degree and the denominator degree are the same , y=The HA= y= * When the power of the numerator is bigger than the denominator, the asymptote with be a slant, or oblique. To find the horizontal asymptote of these equations you have to use long division.

7. X-intercepts. To find the x- intercepts you have to take the numerator and set it equal to zero. Going back to the first equation this is what find the x-ints, would look like. The you solve for x by dividing both sides by -5 . So you would get, and since zero can’t be divided , There are no x- intercepts.

8. Y- intercepts. To find the y- intercepts you set x = to zero. And it equals 0, so the y- intercept would be at the origin, (0,0).

9. tablesTo find possible points of where the line would cross on a graph, you make an x-y tables, of x’s to the left and right of each asymptote.

10. Domain and RangeThe domain of a function is the set of all possible x values which will make the function "work" and will output real y-valuesThe range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.

11. The domain in the graph is (-∞,o)U (0,∞)The range would be the same.

12. Domain and Range cont.The Domain of this function would beD: (-∞,-3] U [-3,1] U [1,∞)The Range would be (-∞,∞)