Transcript 5.2 Graph Simple Rational Functions

Factoring Practice
1. x2 – 16
(x – 4)(x + 4)
2. x3 + 27
(x + 3)(x2 - 3x + 9)
3. 25x2 + 15
5(5x2 + 3)
4. x2 – 10x + 24
(x – 6)(x – 4)
5. 16x2 -36
4(2x – 3)(2x + 3)
6. 27x3 - 8
(3x – 2)(9x2 +6x + 4)
5.2 Graphing Simple Rational
Functions
p. 310
What 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.
p ( x)
f ( x) 
q ( x)
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)
• One form:
a
y
k
xh
• Has 2 asymptotes: x=h (vert.) and y=k (horiz.)
• Graph 2 points on either side of the vertical
asymptote.
• Draw the branches.
Hyperbola (continued)
• Second form:
ax  b
y
cx  d
• Vertical asymptote: Set the denominator
equal to 0 and solve for x.
• Horizontal asymptote:
a
y
c
• Graph 2 points on either side of the vertical
asymptote. Draw the 2 branches.
6
Graph the function y = x . Compare the
graph with the graph of y = 1
x .
SOLUTION
STEP 1
Draw the asymptotes x = 0 and y = 0.
STEP 2
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 3
Draw the branches of the hyperbola so
that they pass through the plotted
points and approach the asymptotes.
6
The graph of y = x lies farther from the
1
axes than the graph of y =
.
x
Both graphs lie in the first and third
quadrants and have the same
asymptotes, domain, and range.
3
y
2
x 1
Ex: Graph
State the domain & range.
Vertical Asymptote: x=1
Horizontal Asymptote: y=2
x
y
-5 1.5
-2
1
2
5
4
3
Left of
vert.
asymp.
Right of
vert.
asymp.
Domain: all real #’s except 1.
Range: all real #’s except 2.
x2
y
3x  3
Ex: Graph
State domain & range.
Vertical asymptote:
3x+3=0 (set denominator =0)
3x=-3
x
x= -1
Horizontal Asymptote: -3
y
a
c
1
y 
3
y
.83
-2 1.33
0
-.67
2
0
Domain: All real #’s
except -1.
Range: All real #’s
except 1/3.
3-D Modeling
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.
• 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 1
Write a function. Let c be the average cost and
m be the number of models printed.
Unit cost • Number printed + Cost of printer
c=
Number printed
300m + 24,000
=
m
STEP 2
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.
STEP 3
Interpret the graph. As more models are
printed, the average cost per model
approaches $300.
Graph the function. State the domain and range.
x
–
1
4. y = 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?
y
a
k
xh
• 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.
Assignment
p. 313,
6-8, 14-20, 28-31