8-3 Graphing Rational Functions

1. UNIT 4: rational functions
8-3 Graphing Rational functions

2. Rational Functions
• Define – Rational Function: is a function with two polynomials
(one in the numerator and one in the denominator)
• Define- point of discontinuity: Value that makes the
denominator zero. (holes / asymptotes)
• Hole: point of discontinuity that can be removed (cancelled out
with the numerator)
• Vertical asymptote: point of discontinuity that can not be
removed (doesn’t cancel with numerator)
• Horizontal asymptote: determine by the degree of numerator
and denominator. (more on that later)

3. Graphing rational functions
• When graphing rational functions you must find points of
discontinuity (holes / asymptotes)
1st -Factor numerator and denominator
2nd – determine points of discontinuity
3rd – graph by making a table
( x − 4)( x + 1)
x+3 x+3 x 2 − 3x − 4
f ( x) = y= 2 g ( x) =
x −5 x − 4x + 3 x−4
( x − 3)( x − 1)
V.A. when: x=5
V.A. when x=1 & x=3 Hole when x=4
x y x y x y
-3 0 -3 0 0 1
1 -1 0 1
2 3
7 5 2 -5
5 6
9 3 5 1
6 7

4. Practice
x+6 x 2 − 3x + 2 x−3
y= f ( x) = g ( x) =
x+4 x−2 3 x 2 − 11x + 6
V.A. x=-4 Hole: x=2 V.A. x=2/3
Hole: x=3
x −5 x+2 2x
y= f ( x) = 2 g ( x) =
x x − x − 12 3x − 1
V.A. x=0 V.A. x=-3 V.A. x=1/3
V.A. x=4

5. Horizontal asymptotes
• To find horizontal asymptotes compare the degree of the
numerator “M” to the degree of the denominator “N”
• If M < N, then y=0 is horizontal asymptote
• If M > N, then No horizontal asymptote
• If M=N, then divide leading coefficients

6. Horizontal asymptotes
• Determine the horizontal asymptotes
2x x+3 x 2 − 3x − 4
f ( x) = y= g ( x) =
x −5 ( x − 3)( x − 1) x−4
M=1 M=1 M=2
N=1 N=2 N=1
H.A. y=2 H.A. y=0 NO H.A.

7. Practice V.A. Holes H.A
• Pg 521 # 17-22 23-28

8. Word problem
• You earn a 75% on the first test of the quarter
how many consecutive 100% test scores do you
need to bring your test average up to a 95%?
75 + 100 x
Write a rational function. Answer: f ( x) =
x +1
Find when the rational
function will be 95% You will need to make
100% on the next 4 test to
bring your test average
up to a 95%

9. Word problem
• A Basketball player have made 5 out of the last 7
free throws. How many more consecutive free
throws do they need to make to have an average
of 80%? 5+ x
Answer: f ( x) =
Write a rational function. 7+ x
Find when the rational You will need to make the
function will be 80% next 3 free throws for
average to be an 80%

10. Practice word problems
• Pg 521 #39,40

11. Word problem
• The function below gives the concentration of
the saline solution after adding x milliliters of
0.5% solution to 100 milliliters of 2% solution.
100(0.02) + x(0.005)
y=
100 + x
• How many ML of the 0.5% solution must be
added to have a combined concentration of
0.9%?
Answer: (search table) X=275