Download - Factoring Practice
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