Definition: A rational function is a function that can be written where p(x) and q(x) are polynomials. 8) Graph Steps to graphing a rational function. 1) Factor first 2) Find the domain. 3) Find any holes 4) Identify the vertical asymptote(s). Include any multiplicity. 5) Find the horizontal asymptote and check to see if the graph crosses it. 6) Find the y-intercept. 7) Find the x-intercept(s)

Domain The domain is all real numbers except the values of x that make the denominator equal to zero. Write in interval notation. Holes If there are nonconstant factors that cancel you will have a hole. To find the location of the hole, set the common factor (the one the you canceled) equal to zero and solve for x. This is the x-coordinate of the hole. Substitute the x-coordinate into the simplified form of the function to find the y-coordinate. Write the hole as a point. Example 1

Vertical Asymptote After you have found any holes, the values of x that make the denominator zero are the location of your vertical asymptotes. The graph cannot cross a vertical asymptote. Vertical asymptotes are vertical lines and should be written as x = Vertical asymptotes can have multiplicity. For single factors the graph goes up on one side of the asymptote and down on the other side For double factors the graph in the same direction on both sides of the asymptote. or Example 1

To find the horizontal asymptote compare the degree of the numerator with the degree of the denominator. There are three possibilities. If the degree of the numerator is larger than the degree of the denominator there is no horizontal asymptote. If the degree of the numerator is smaller than the degree of the denominator the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator the horizontal asymptote is the ratio of the leading coefficient of the numerator to the leading coefficient of the denominator. For example: if the horizontal asymptote would be Horizontal asymptotes are horizontal lines and should be written as y = Horizontal Asymptote Horizontal asymptotes deal with the end behavior of the graph. That is what is happening as x gets really large and really small.

Horizontal Asymptote (continued) Horizontal asymptotes may be crossed. To find if and where the graph crosses the horizontal asymptote set the function equal to the y-value of the asymptote and solve for x If the point where the graph crosses the horizontal asymptote is not between two vertical asymptotes, it will come back toward the asymptote once it crosses it. For example: Example 1

Y-intercept To find the y-intercept let x= 0 and solve for y. X-intercept To find the x-intercept let y= 0 and solve for x. Write as a point. Note: Since the denominator can never equal zero, you really only need to set the numerator equal to zero. Example 1

Example 1: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ (back to notes) (- ,-2)  (-2, 2)  (2,  ) x = -2 y = 2 does not cross (back to notes)

Example 2: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ None

Example 3: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ None

Example 4: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ None Crosses at x = 4

Example 5: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ None

Example 6: D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________ None

D: ______________________ Hole: ____________________ VA: ____________________ HA: _____________________ y-int: ____________________ x-int: _____________________