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Name:_______________________ Date assigned:______________ Band:________

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 functionmightswitch signs at x-intercepts, they dont 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 Hole 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(2 5)( 1)

( )( 1)

x xp x

x

.

(a) What would this function look like when graphed?

(b) There is a hole on this graph. What is the height of the hole?

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Section 1: From a graph to a sign analysis, and vice versa

1. Draw a sign analysis for the followinggraph

2. A function has the sign analysis below and a horizontal asymptote at 1y . Come up with a potential sketch!(Hint: Draw the VA and HA dashed lines first! That helps!)

x

y

x

y

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3. A function has the sign analysis below and a horizontal asymptote at 3y . Come up with a potential sketch!(Hint: Draw the VA and HA dashed lines first!)

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!

x

y

x

y

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Section 2: Graphing Rational Functions from Scratch

1. Analyze

2

2

( 2)( 3)( )

( 1) ( 3)

x x xf x

x x

(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, whats 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!

x

y

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Section 3: Problems

Come up with a decent sketch of the rational functions below. Check your graphs on Geogebra or your graphing

calculator. Both wont show you holes, however. Ah, computers. Inferior to the human mind.

(1) 22

1( )

2 3

xf x

x x

(2) 2( )1

xg x

x

x

y

x

y

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(3) 3 3( )2 4

xR x

x

(4)2

1

9

xy

x

x

y

x

y

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(5) 22

12

4

x xy

x

(6) 25 4( )5

x xk x

x

x

y

x

y

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(7) 2 4 12( )4

x xf x

x

(8)3 9

xy

x x

x

y

x

y

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(9) 21

( )1

xg x

x

(10) 2 1xyx

x

y

x

y

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(11) 1y xx

[hint: rewrite]

(12) 42

1( )

4

xf x

x

x

y

x

y