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Agenda• Don't forget about resources on
mrwaddell.net
• Chapter 8: Rational Functions• Can you graph simple rational functions?
Vocabulary
Domain
Range
The places along the x axis where the graph lives
The places along the y axis where the graph lives
Vocabulary
Rational Expression
Asymptote
Vertical
Horizontal
A quotient of 2 polynomials
An invisible line that the graph gets close to, but never touches.
Vocabulary
Vertical Asymptote Occurs anywhere the fraction is divided by zero.
You must, MUST, MUST check to find out where the denominator equals zero.
This is NOT optional.
Checking for Vertical Asymptotes
x
x
6
62 3 Is this a Rational function?
Yes, the numerator is cubic, the denominator is linear
Where does the denominator equal zero?
6x Take the bottom= 0Set it equal to zero and solve.
x = 0
There is a vertical asymptote at x = 0
Checking for Vertical Asymptotes
732
2852
23
xx
xx x
Is this a Rational function?
No, the 2x messes it up!
Finding the domain
126
62 3
x
x What is the domain of the expression?
Where does the denominator equal zero?
6x -12 Take the bottom= 0Set it equal to zero and solve.
6x = 12
There is a vertical asymptote at x = 2
x = 2
The domain is: All Reals except x = 2
Graphing Simple Rational Functions
xy
1 The parent function for all
rationals.
Where does the denominator equal zero?
x Take the bottom= 0Set it equal to zero and solve.
There is a vertical asymptote at x = 0
The domain is: All Reals except x = 0
xy
1
x= 0 is the vertical asymptote, and y=0 is the horizontal asymptote
The domain is all reals and x ≠ 0
The range is all reals and y ≠ 0
xy
6
x= 0 is the vertical asymptote and y =0 is the horizontal asymptote.
The domain is all reals except x = 0
The range is all reals except y = 0
xy12
x= 0 is the vertical asymptote and y = 0 is the horizontal asymptote.
The domain is all reals except x = 0
The range is all reals except y = 0
xy
4
x= 0 is the vertical asymptote and y = 0 is the horizontal asymptote.
The domain is all reals except x = 0
The range is all reals except x = 0
Graphing Simple Rational Functions
x
ay What does the “a” do to the
equation?
How did the graphs change as “a” went from 1 to 12?
Just like all the other translations, “a” makes it skinny or fat, and a negative “flipped it” around the X AXIS.
Graphing Simple Rational Functions
khx
ay
What do you think the “h” and
the “k” do to the graph?
If you said, “The same thing as with radicals and quadratics and polynomials,” you are RIGHT!
“h” moves it left and right, (but opposite to the sign” and “k” moves it up and down!
3
2
x
y
x= -3 is the vertical asymptote and y = 0 is the horizontal asymptote.
The domain is all reals except x = -3
The range is all reals except x = 0
43
2
x
y
x= -3 is the vertical asymptote and y = 4 is the horizontal asymptote.
The domain is all reals except x = -3
The range is all reals except x = 4
Graphing Rational Functions
dcx
baxy
Now they are a little more complex.
The numerator and denominator are both lines, that is very important.
The vertical asymptote is at x = -d/c
The horizontal asymptote is at y = a/c
4
32
x
xy
x = 4 is the vertical asymptotey=2 is the horizontal asymptote.
The domain is all reals except x = 4
The range is all reals except y = 2/1 = 2