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