Transcript 8. Graphing Simple Rational Functions

Section 9.2
Graphing Simple
Rational Functions
Basic Curve
1
What does y 
look like?
x
x
y
-8
-4
-2
-1
-0.5
0.5
1
2
4
8
-0.125
-0.25
-0.5
-1
-2
2
1
0.5
0.25
0.125
y
10
8
6
4
2
x
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Let’s look at some graphs!
1
y
x
3
y
x
5
y
x
As the number on top
becomes a larger positive, the
branches in quadrants I and
III widen.
y
10
8
6
4
2
x
-10
-8
-6
-4
-2
-2
2
4
6
8
10
-4
-6
Each of the two pieces of the
curve is called a ‘branch.’
-8
-10
The curve itself (both branches) is called a ‘hyperbola.’
Let’s look at more graphs!
1
3
5
y
y
y
x
x
x
As the number on top
becomes a larger negative,
the branches in quadrants II
and IV widen.
y
10
8
6
4
2
x
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Let’s look at even more!
3
y
x3
3
y
x4
y
10
8
6
Notice that the number on the
bottom will translate the graphs y = 0
left or right (in the opposite
direction).
Notice that the branches never
touch the dotted line. The dotted
x = -4
line the branches never touch is
called an ‘asymptote.’
-10
-8
-6
-4
-2
4
2
-2
x
2
4
-4
-6
-8
-10
x=3
6
8
10
How ‘bout a few more!
3
y  3
x
3
y  4
x
y
10
8
6
Notice that the number on the
right will translate the graphs
up or down (in that direction).
y=4
4
2
x
-10
y = -3
-8
-6
-4
-2
-2
-4
-6
-8
-10
x=0
2
4
6
8
10
x: The domain is all the possible values that are
allowed to go into the x.
Let’s talk Domain!
1
y
x
Recall: You can’t divide by 0!
So what value of x would send in the math
police?
Therefore, x is allowed to be any real number, except zero.
Notation (D: All Real Numbers, but x = 0.)
3
y
x3
So what value of x would make the
denominator zero?
Therefore, x is allowed to be any real number, except three.
Notation (D: All Real Numbers, but x = 3.)
y: The range is all the possible values that are
allowed to come out of a function.
Let’s talk Range!
1
y
x
Since x cannot be 0, the expression 1/x
cannot be 0. Therefore, y cannot be 0.
Notation (R: All Real Numbers, but y = 0.)
3
y  3
x
Since x cannot be 0, the expression 3/x
cannot be 0. Therefore, y cannot be 0 – 3,
which is –3.
Notation (R: All Real Numbers, but y = -3.)
Graph and state the
domain and the range!
3
y
5
x6
y
10
8
y=5
6
4
•We know we have a hyperbola.
•How many units left/right?
•How many units up/down?
•What value can x not have?
•What value can y not have?
2
x
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
x = -6
2
4
6
8
10
Graph and state the
domain and the range!
3
y
2
x4
•We know we have a hyperbola.
•How many units left/right?
•How many units up/down?
•What value can x not have?
•What value can y not have?
y
10
8
6
4
y=2
2
x
-10
-8
-6
-4
-2
-2
2
-4
-6
-8
-10
x=4
4
6
8
10