Applications
A population P after t years is modeled by:
400038000900)(
t
ttP
a) What does P(0) represent ?
b) What is the horizontal asymptote?
c) What does the horizontal asymptote represent ?
Warm-up
Determine all asymptotes for each function.
1.
2.
4252)( 2
2
x
xxxf
xxxf 1)(
2
Two key things to REMEMBER…
1. A graph will only cross the x-axis
at x-intercepts (zeros)!
2. A graph can not cross a vertical
asymptote!
3.4 Graph of a Rational Function
A. Sketch the graph of: a) Domain b) Reduce to lowest termsc) y-intercept: Evaluate f(0)d) x-intercepts: Zeros of numerator
Multiplicity: Even– touches ; Odd– crossese) VA : zeros of denominator (use reduced function)
Multiplicity: Even (y values stay on same side of x-axis) Odd (y values switch sides)
f) Holes: terms which cancelled.g) HA/OA: End Behavior h) Test points each side of x=intercepts and VAs.i) Check: Does your graph pass the vertical line test?
422)( 2
2
xxxxxf
Analyzing Rational Graphs
32)( 2
2
xxxxf
a) Domain
b) Reduce to lowest terms
c) y-intercept:
d) x-intercepts: (and Multiplicity)
e) VA : (and Multiplicity)
f) Holes:
g) HA/OA:
h) Test points .
i) Check:
Handout: 3.4 Graphing Rational Functions
9642)( 1) 2
2
x
xxxR
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
Analyzing Rational Graphs
x-int: (-1,0) y-int: (0,2/3)No symmetry Hole: (3,4/3)VA: x= -3HA: y = 2
9642)( 2
2
x
xxxR
Analyzing Rational Graphs
x-int: (-1,0) y-int: (0,2/3)No symmetry Hole: (3,4/3)VA: x= -3HA: y = 2
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
43)( 2) 2
x
xf
3.4 Handout
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
Analyzing Rational Graphs
f (x) 3
x 2 4
3(x 2)(x 2)Analyze the graph of:
x-int: None y-int: (0,-3/4)y-Axis SymmetryD: {x | x -2, 2 }
3.4 Handout
123)( 3)
2
x
xxxf
Analyzing Rational Graphs
f (x) x 2 3x 2
x 1
(x 2)(x 1)x 1
Analyze the graph of:
x-int: (-2,0) (-1,0) y-int: (0,-2)No SymmetryD: {x | x 1 }
3.4 Handout
4582)( 4) 2
23
xx
xxxxf
B. Build a rational function from information
Recall for Polynomials…• when c is a zero • (x-c) is a factor
Build a Rational Function with the following properties
1. VA at x= -2 and x= 2 , HA at y = 2, x-intercepts at -3, 3
2. VA at x= 3 , HA at y = 0, no x-intercepts
B. Find the equation from the graph
Definition: Multiplicity of Vertical Asymptotes.
21
x
y2)2(
1
x
y
3)2(1
x
y 4)2(1
x
y
ODD multiplicity of VA
y-values change sign on each side of VA
y-values do not change sign on each side of VA
EVEN multiplicity of VA
Handout 3.4 #5)
Handout 3.4 #6
Handout 3.4 #7
1. Direct Variationy varies directly with x is modeled by:
k is the constant of proportionality
kxy
2. Inverse Variationy varies inversely as x, if there is a constant k such that:
xky
3. Joint Variation Problems
y varies jointly with if there is a constant k such that:
nxxx ,...,, 21
nxxxky 21
4. Solving Variation Problems1) Set up equation2) Use known information to find k3) Plug k back into 1)4) Answer question using equation in 3)
Solving inequalities – true/false quizTrue or False.
1. The solution set of is
2. The inequality can be solved by multiplying
both sides by , resulting in the equivalent inequality
162 x ),4(
232
xx
)3(22 xx
3x