Algebra II Graphing Rational Functions Notes
Name: _______________________
Holes:

Definition: ____________________________________________________________
Identifying holes, vertical asymptotes, and horizontal asymptotes from an equation:
1. Always begin with finding horizontal asymptotes. You do this by comparing the
________________________ of the numerator and denominator.

If the degree of the numerator is LESS than the degree of the denominator: the HA is always
_______________. Example: y 

x 1
.
3  x2
HA:______________
If the degree of the numerator is EQUAL to the degree of the denominator: the HA is the
_____________________________________. Example: y 

3x 2  2 x
4 x2  1
HA:__________
If the degree of the numerator is GREATER than the degree of the denominator: the HA is
__________________. Example: y 
2 x  10
5
HA:_______________
2. Factor the problem like you are going to simplify it. Example:
y
x2  4

x2  5x  6
3. Identify the holes. You do this by _________________ like factors. Set the factor that cancels equal
to ____________ and solve.
4. Identify the vertical asymptotes. You do this AFTER you find the holes. Set the remaining factors
from the denominator ______________________________________________.
Algebra II Graphing Rational Functions Notes
Name: _______________________
Characteristics of Functions
1. Domain: The domain of every rational function is ________________________ except the
______________ asymptote(s).
2. Range: The range of every rational function is _______________________ except the
______________ asymptote(s).
3. Zeros (x-intercepts): The point or points where the function touches the _________________. The y
value at each x-intercept is ___________.

Calculator: Go to calc menu and choose zero. The calculator will ask 3 questions.
Step 1: put the cursor on top of the x-intercept
Step 2: LEFT BOUND? push left twice and hit enter
Step 3: RIGHT BOUND? push right four times and hit enter
Step 4: GUESS? put the cursor on top of the x-intercept and hit enter
4. y-intercept: The point where the function touches the ___________________. The x value at each
y-intercept is ____________.

Calculator: Go to the calc menu and choose value. The calculator says X=. Type zero and hit
enter.
5. Maximum(s): A point that is _____________ than the nearby points. This is a point so your answers
should always be (x, y). Many rational functions do not have maximums so your answers may be
none.
6. Minimum(s): A point that is ____________ than the nearby points. This is a point so your answers
should always be (x, y). Many rational functions do not have minimums so your answers may be
none.

Using a calculator to find a max or a min: Go to calc menu and choose max or min. The
calculator will ask 3 questions.
Step 1: put the cursor on top of the max or min
Step 2: LEFT BOUND? Push left twice and hit enter
Step 3: RIGHT BOUND? Push right four times and hit enter
Step 4: GUESS? put the cursor on top of the x-intercept and hit enter
7. Increasing interval: This is where the function is __________________. It means that the y is
increasing so you should answer with x values. Answers should look like _____ < x < _____.

For rational functions, you will often use the vertical asymptote and either  or  .
Algebra II Graphing Rational Functions Notes
Name: _______________________
8. Decreasing interval: This is where the function is __________________. It means that the y is
decreasing so you should answer with x values. Answers should look like _____ < x < _____.

For rational functions, you will often use the vertical asymptote and either  or  .
9. End Behavior:

As x   (the graph goes right), what is y doing?

As x   (the graph goes left), what is y doing?

There are only three possible answers to either question.
i.
y   (the graph goes up)
ii.
y   (the graph goes down)
iii.
y  some number (the graph becomes a flat, horizontal line).