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Precalculus | Packer Collegiate Institute
Rational Functions #4
Warm Up 1: SKETCHES
A sign analysis tells you where a function is positive or negative. The only ways a function can switch from being positive
to being negative (or being negative to being positive) are at x-intercepts, vertical asymptotes, and holes. However, we
have seen that just because a function might switch signs at x-intercepts, they don’t have to switch signs.
Below you are going to draw quick sketches of examples of functions with the following properties…
x-intercept at x=2
VA at x=2
Switches
from – to +
at x=2
Switches
from + to –
at x=2
Stays –
at x=2
Stays +
at x=2
Warm Up 2: I have the function p( x)
(2 x 5)( x 1)
.
( x 1)
(a) What would this function look like when graphed?
(b) There is a hole on this graph. What is the height of the hole?
1
Hole at x=2
Section 1: From a graph to a sign analysis, and vice versa…
1. Draw a sign analysis for the following
graph…
y
x
2. A function has the sign analysis below and a horizontal asymptote at y 1 . Come up with a potential sketch!
(Hint: Draw the VA and HA dashed lines first! That helps!)
y
2
x
3. A function has the sign analysis below and a horizontal asymptote at y 3 . Come up with a potential sketch!
(Hint: Draw the VA and HA dashed lines first!)
y
x
4. A function has the sign analysis below and no horizontal asymptote.
(a) What is the end behavior?
(b) Come up with a potential sketch!
y
x
3
Section 2: Graphing Rational Functions from Scratch
x 2 ( x 2)( x 3)
1. Analyze f ( x)
( x 1)2 ( x 3)
(a) Find the holes, VAs, and x-intercepts. Also, answer what the domain of the rational function is.
(b) Determine the end behavior (Is there a HA? If not, what’s the graph going to be doing for large positive and
negative values?)
(c) Do a sign analysis of the function!
(d) Do a quick sketch!
y
4
x
Section 3: Problems
Come up with a decent sketch of the rational functions below. Check your graphs on Geogebra or your graphing
calculator. Both won’t show you holes, however. Ah, computers. Inferior to the human mind.
(1) f ( x)
x2 1
x2 2 x 3
y
x
(2) g ( x )
x2
x 1
y
5
x
(3) R ( x )
3x 3
2x 4
y
x
(4) y
x 1
x2 9
y
6
x
(5) y
x 2 x 12
x2 4
y
x
(6) k ( x)
5 4x x2
x5
y
7
x
(7) f ( x)
x 2 4 x 12
x4
y
x
(8) y
x
x 9x
3
y
8
x
g ( x)
(9)
x 1
x2 1
y
x
(10) y
x2 1
x
y
9
x
(11) y x
1
[hint: rewrite]
x
y
x
(12) f ( x)
x4 1
x2 4
y
10
x
0
