Graphing Rational Functions
Download
Report
Transcript Graphing Rational Functions
ADV122
GRAPHING RATIONAL FUNCTIONS
Warm Up
Graph the function
𝑓 𝑥 =− 𝑥+3
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
ADV122
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.
ADV122
GRAPHING RATIONAL FUNCTIONS
Asymptotes
Places on the graph the function will approach,
but will never touch.
ADV122
GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
x
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
No horizontal shift.
No vertical shift.
A HYPERBOLA!!
ADV122
GRAPHING RATIONAL FUNCTIONS
W𝐡𝐚𝐭 𝐝𝐨𝐞𝐬 𝒇 𝒙 =
𝟏
−
𝒙
look like?
ADV122
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
ADV122
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 = 0
ADV122
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 = 1
ADV122
GRAPHING RATIONAL FUNCTIONS
You try!!
1
1. 𝑦 =
+2
𝑥
2. 𝑦 =
1
𝑥+3
−4
ADV122
GRAPHING RATIONAL FUNCTIONS
How do we find asymptotes
based on an equation only?
ADV122
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.
ADV122
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
ADV122
GRAPHING RATIONAL FUNCTIONS
3 cases
ADV122
GRAPHING RATIONAL FUNCTIONS
If the degree of the denominator
is GREATER than the
numerator.
The Asymptote is y=0 ( the x-axis)
ADV122
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.
ADV122
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