Graphing Rational Functions:
There are many steps when graphing rational expressions. Luckily you are familiar with most of the steps to do this! Let's put them all together.
Follow along with the completed problem on your worksheet as you read these steps.
Follow along with the completed problem on your worksheet as you read these steps.
- ALWAYS FACTOR FIRST
- Simplify the expression - cancel common factors (if applicable)
- Identify the y-intercept. You do this by setting x = 0 and solving for y. Remember, this is a coordinate point!
- Identify the x-intercept. You do this by setting the numerator = 0 and solving for x. Remember that this is a coordinate point! **Must do this after you simplify the expression**
- Vertical Asymptote: This is the value of x that makes your function undefined. Set your denominator = 0 and solve for x. This will be a vertical line in the form x = c (where c is the value you solved for)
- Horizontal Asymptote: Look at the degree of the numerator and denominator. (Of the simplified expression). **Remember - the degree of a constant is 0.** This will be a horizontal line in the form y = c (where c is one of the 2 things below)
Simplified Function: Horizontal Asymptote:
7. Hole: This comes from the cancelled factor. The hole is a coordinate point.
x value of the hole - set the cancelled factor = 0 and solve
y values of the hole - plug in the x value you found
to the SIMPLIFIED function and solve.
8. Sketch a graph. Include all of the things that you found.
If you want to pick extra points on the graph, you can make a table.
9. Domain: List all of the values in interval notation, reading the graph from left to right. Exclude any vertical asymptote values or x-coordinate of the hole.
10. Range: List all of the values in interval notation, reading the graph from bottom
to top. Exclude any y values of the hole or horizontal asymptotes.
11. Limits: Describe the end behavior using limit notation as well as the vertical asymptote behavior. Keep in mind that you need to describe what happens as the function approaches the vertical asymptote from both sides.
x value of the hole - set the cancelled factor = 0 and solve
y values of the hole - plug in the x value you found
to the SIMPLIFIED function and solve.
8. Sketch a graph. Include all of the things that you found.
If you want to pick extra points on the graph, you can make a table.
9. Domain: List all of the values in interval notation, reading the graph from left to right. Exclude any vertical asymptote values or x-coordinate of the hole.
10. Range: List all of the values in interval notation, reading the graph from bottom
to top. Exclude any y values of the hole or horizontal asymptotes.
11. Limits: Describe the end behavior using limit notation as well as the vertical asymptote behavior. Keep in mind that you need to describe what happens as the function approaches the vertical asymptote from both sides.