Name:
Date:
Advanced Math
Test Review: Rational Functions
Outline:
1) Given a rational function, be able to find:
a) x – intercepts
b) y – intercept
c) coordinates of holes
d) horizontal/slant asymptotes (aka - end behavior asymptotes)
f) vertical asymptotes
2) You should be able to sketch a graph of a rational function using the information found above and a
few exact points.
3) Be able to write a function formula for a rational function given a graph.
ax + b
4) Be able to turn a function from the form f (x) =
into the transformation form
cx + d
a
1
f (x) =
+ k . Then, identify the transformations from the parent function g(x) = to f (x).
x -h
x
Order is important - always put the vertical shift last.
5) Be able to solve equations with rational expressions.
6) Be able to add rational expressions.
Notes to help you study:
Finding end behavior asymptotes:
 If degree is higher in the numerator, use long division to find the asymptote.
 If degree is higher in the denominator, there is a horizontal asymptote at y = 0.
 If degree is the same in the numerator and denominator, divide the coefficients of the leading
terms to find the horizontal asymptote.
Finding x - intercepts: set numerator = 0 and solve for x.
Finding y – intercept: plug in 0 for x.
Finding vertical asymptotes: they are the zeros of the denominator unless they are a hole as described
below:
Finding holes:
 If factors in numerator and denominator cancel, there is a hole at the x value that makes that
factor = 0. (to find the y – coordinate of that hole, plug x into the function AFTER you cancel
out the factors)
1
Name:
Date:
Advanced Math
PRACTICE:
Simplify the following.
1.
2
x
- 2
x -1 x - 2x +1
2
2.
1
4
3 +
3
3x 6x + 3x 2
4.
-6x
2
2x
+
=
x + 5x + 4 x +1 x + 4
Solve for x.
3.
5
2
+
=1
2x 3(x +1)
5. Analyze and graph the function f ( x) 
2
( x  3)
. Include the x-intercepts, y – intercept, holes,
( x  3)( x  4)
and all asymptotes
2
Name:
Date:
Advanced Math
6. Analyze each function below. Identify the x-intercepts, y – intercept, holes, and all asymptotes.
a. f ( x) =
( x - 2)( x + 3)
( x 3 + 3x 2 - x - 3)
b. f ( x) =
2x 2 + 7x - 4
x2 + x - 2
c. f ( x) 
x2  2
x 1
Hint: factor the denominator using grouping)
3
Name:
Date:
Advanced Math
7. Given f ( x ) =
4x + 9
x+3
a. Express f (x )in transformation form. (Divide). ( f (x) =
b. Identify the transformations to go from g(x) =
a
+ k)
x -h
1
to f (x ) in the proper order.
x
c. Identify the domain
d. Identify the range
e. Write the equation for the vertical asymptote
f. Write the equation for the horizontal asymptote.
g. Find the x – intercept.
h. Find the y – intercept.
8.
Write the equation of the rational function graphed below. Assume all asymptotes and zeros are
 3
at integers. A point on the function is 1,

 2 
9. Assume that the graph in question 8 above has a hole where x = 2.
a. Now what is its equation?
b. What is the y-coordinate of the hole?
4
5