Factoring Practice1. x2 – 16

2. x3 + 27

3. 25x2 + 15

4. x2 – 10x + 24

5. 16x2 -36

6. 27x3 - 8

(x – 4)(x + 4)

(x + 3)(x2 - 3x + 9)

5(5x2 + 3)

(x – 6)(x – 4)

4(2x – 3)(2x + 3)

(3x – 2)(9x2 +6x + 4)

5.2 Graphing Simple Rational Functions

p. 310What is the general form of a rational function?

What does the h & k tell you?What does the graph of a hyperbola look like?What does the graph of ax+b/cx+d tell you?

What information does the domain & range tell you?

Rational Function

• A function of the form

where p(x) & q(x) are polynomials and q(x)≠0.

)(

)()(

xq

xpxf

Hyperbola

• A type of rational function.

• Has 1 vertical asymptote and 1 horizontal asymptote.

• Has 2 parts called branches. (blue parts) They are symmetrical.

We’ll discuss 2 different forms.

x=0

y=0

Hyperbola (continued)Hyperbola (continued)

• One form:

• Has 2 asymptotes: x=h (vert.) and y=k (horiz.)

• Graph 2 points on either side of the vertical asymptote.

• Draw the branches.

khx

ay

Hyperbola (continued)

• Second form:

• Vertical asymptote: Set the denominator equal to 0 and solve for x.

• Horizontal asymptote:

• Graph 2 points on either side of the vertical asymptote. Draw the 2 branches.

dcx

baxy

c

ay

Graph the function y = . Compare the graph with the graph of y = .1

x

6x

SOLUTION

STEP 1Draw the asymptotes x = 0 and y = 0.

Plot points to the left and to the right of the vertical asymptote, such as (–3, –2), (–2, –3), (2, 3), and (3, 2).

STEP 2

Draw the branches of the hyperbola so that they pass through the plotted points and approach the asymptotes.

STEP 3

The graph of y = lies farther from the axes than the graph of y = .

6x 1

x

Both graphs lie in the first and third quadrants and have the same asymptotes, domain, and range.

Ex: Graph State the domain & range.

21

3

x

y

Vertical Asymptote: x=1

Horizontal Asymptote: y=2

x y

-5 1.5

-2 1

2 5

4 3Domain: all real #’s except 1.

Range: all real #’s except 2.

Left of vert.

asymp.

Right of vert.

asymp.

Ex: GraphState domain & range.Vertical asymptote:3x+3=0 (set denominator =0)

3x=-3

x= -1

Horizontal Asymptote:

c

ay

3

1y

x y

-3 .83

-2 1.33

0 -.67

2 0

Domain: All real #’s except -1.

Range: All real #’s except 1/3.

33

2

x

xy

A 3-D printer builds up layers of material to make three dimensional models. Each deposited layer bonds to the layer below it. A company decides to make small display models of engine components using a 3-D printer. The printer costs $24,000. The material for each model costs $300.

3-D Modeling

• Write an equation that gives the average cost per model as a function of the number of models printed.

• Graph the function. Use the graph to estimate how many models must be printed for the average cost per model to fall to $700.

• What happens to the average cost as more models are printed?

SOLUTION

STEP 1Write a function. Let c be the average cost and m be the number of models printed.

c =Unit cost • Number printed + Cost of printer

Number printed

=300m + 24,000

m

STEP 2

STEP 3

Interpret the graph. As more models are printed, the average cost per model approaches $300.

Graph the function. The asymptotes are the lines m = 0 and c = 300. The average cost falls to $700 per model after 60 models are printed.

Graph the function. State the domain and range.

4. y =x – 1 x + 3

SOLUTION

ANSWER domain: all real numbers except – 3, range: all real numbers except 2.

• What is the general form of a rational function?

• What does the h & k tell you?

Asymptotes are x = h, y = k• What does the graph of a hyperbola look like?

Two symmetrical branches in opposite quadrants.• What does the graph of ax+b/cx+d tell you?

cx+d = 0 is the vertical asymptote and y = a/c is the horizontal asymptote

• What information does the domain & range tell you?

Domain tells what numbers can be used for x and the range is the y numbers when put into the equation.

khx

ay

Assignment

p. 313,

6-8, 14-20, 28-31