ADV122
GRAPHING RATIONAL FUNCTIONS
Warm Up
Graph the functions
𝑓 𝑥 =− 𝑥+3
f ( x)  x  3  2
f ( x)  3 x  1
2
+4
ADV122
GRAPHING RATIONAL FUNCTIONS
We have graphed several functions,
now we are adding one more to the
list!
Graphing Rational Functions
ADV122
GRAPHING RATIONAL FUNCTIONS
Parent Function: 𝒇 𝒙 =
𝟏
𝒙
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GRAPHING RATIONAL FUNCTIONS
Pay attention to the transformation clues!
(-a indicates a reflection
in the x-axis)
a
f(x) =
+k
x–h
vertical translation
(-k = down, +k = up)
horizontal translation
(+h = left, -h = right)
Watch the negative sign!! If
h = -2 it will appear as x + 2.
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GRAPHING RATIONAL FUNCTIONS
Asymptotes

Places on the graph the function will approach,
but will never touch.
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GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
x
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
No horizontal shift.
No vertical shift.
A HYPERBOLA!!
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GRAPHING RATIONAL FUNCTIONS
W𝐡𝐚𝐭 𝐝𝐨𝐞𝐬 𝒇 𝒙 =
𝟏
−
𝒙
look like?
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GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
x+4
x + 4 indicates a
shift 4 units left
Vertical Asymptote: x = -4
No vertical shift
Horizontal Asymptote: y = 0
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GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
–3
x+4
x + 4 indicates a
shift 4 units left
Vertical Asymptote: x = -4
–3 indicates a shift 3
units down which
becomes the new
horizontal asymptote
y = -3.
Horizontal Asymptote: y = -3
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GRAPHING RATIONAL FUNCTIONS
Graph: f(x) =
x
+6
x–1
x – 1 indicates a
shift 1 unit right
Vertical Asymptote: x = 1
+6 indicates a shift 6
units up moving the
horizontal asymptote
to y = 6
Horizontal Asymptote: y = 6
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GRAPHING RATIONAL FUNCTIONS
You try!!
1
1. 𝑦 =
+2
𝑥
2. 𝑦 =
1
𝑥+3
−4
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GRAPHING RATIONAL FUNCTIONS
How do we find asymptotes
based on an equation only?
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GRAPHING RATIONAL FUNCTIONS
Vertical Asymptotes (easy one)

Set the denominator equal to zero and solve for
x.

Example: 𝑦 =
6
𝑥−3

x-3=0
x=3

So: 3 is a vertical asymptote.
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GRAPHING RATIONAL FUNCTIONS
Horizontal Asymptotes (H.A)


In order to have a horizontal asymptote, the
degree of the denominator must be the same, or
greater than the degree in the numerator.
Examples:
𝑥 2 −3
 𝑦 =
𝑥+7
𝑥 3 −2
 𝑦 = 3
𝑥 −2
𝑥+1
 𝑦 = 2
𝑥
No H.A because 2 > 1
Has a H.A because 3=3.
Has a H.A because 1 < 2
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GRAPHING RATIONAL FUNCTIONS
The x-intercepts of the graph are the real
zeros of the numerator.
The graph has a vertical asymptote at
each real zero of the denominator.
The graph has at most one horizontal
asymptote.
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GRAPHING RATIONAL FUNCTIONS
If the degree of the denominator
is GREATER than the
numerator.

The Asymptote is y=0 or y=the vertical shift
value
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GRAPHING RATIONAL FUNCTIONS
If the degree of the denominator
and numerator are the same:

Divide the leading coefficient of the numerator
by the leading coefficient of the denominator in
order to find the horizontal asymptote.
6𝑥 3
3𝑥 3 −2

Example: 𝑦 =

Asymptote is 6/3 =2.
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GRAPHING RATIONAL FUNCTIONS
If there is a Vertical Shift


The asymptote will be the same number as the
vertical shift.
(think about why this is based on the examples
we did with graphs)
5
+7
𝑥−3

Example:

Vertical shift is 7, so H.A is at 7.
ADV122
GRAPHING RATIONAL FUNCTIONS
Homework

http://www.kutasoftware.com/FreeWorksheets
/Alg2Worksheets/Graphing%20Simple%20Rati
onal%20Functions.pdf