INVERSE VARIATIONS
Recall:
y
k
x
where k  0
Inverse variation (varies
inversely as) is the situation
that occurs when two
variables x and y are so
related that when one
variable is multiplied by k, the
other is divided by k.
Domain:  x : x  0
Parent Function: y 
where x  0
4
1
where x  0
x
Branch
2
Range:  y : y  0
y0
-2
Branch
-4
Shape: Hyperbola
5
x
y
1
2
8
−
1
4
2
2
-2
-2
4
1
-4
-1
8
1
2
-8
-12
x
x
y
4
1
2
-8
2
-1
-4
-5
5
-2
-4
-6
-8
FAMILIES OF RECIPROCAL FUNCTIONS
𝟏𝟐
𝒙
Families of Reciprocal Functions
8
6
g x  =
12
x
Parent Function
y
1
,x  0
x
Stretch 𝑎 > 1
Compression (Shrink) 0 < 𝑎 < 1
y
a
,x  0
x
4
Domain:  x : x  0
2
Range:  y : y  0
Asymptotes: x  0
What is the graph of 𝑦 = , 𝑥 ≠ 0? Identify the x- and y-intercepts
𝑥
and the asymptotes of the graph. State the domain and range of
the function.
8
1. Make a table of values
6
4
and plot points.
g x  =
2. Identify and graph asymptotes.
3. Connect with a smooth curve.
GOT IT #1 P.508
a.Graph 𝒚 =
x
4
Domain:  x : x  0
Asymptotes: x  0
1
GRAPHING AN INVERSE VARIATION FUNCTION
RECIPROCAL FUNCTIONS
Functions of the form:
f x  =
2
Both x-axis and y-axis
are asymptotes.
Range:  y : y  0
a
y
x
4
-5
y0
5
-2
Reflection 𝑎 < 0 in x-axis
-4
Translations
(Horizontal by h; Vertical by k)
-6
Intercepts: None
-8
All transformations combined
1
 k, x  h
xh
a
y
 k, x  h
xh
y
b. Yes, they have similar graphs
1
GOT IT #2 P.509
GRAPHING A TRANSFORMATION
What is the graph of 𝑦 =
range.
2
+
𝑥−2
3? Identify the domain and
x
y
x
y
6
7
-6
2.75
4
3
5
-2
2.5
4
4
-1
1
6
3.5
0
2
1.5
-1
10 3.25
2
f x  =
1
2
1
1
b. 𝑦 = 𝑥 is a stretch of the graph 𝑦 = 𝑥 by a factor of 2.
8
2.5
1
a. 𝑦 = 2𝑥 is a shrink of the graph 𝑦 = 𝑥 by a factor of 2.
x-2
1
+3
c. 𝑦 = − 2𝑥 is a reflection across the x-axis and a shrink
1
1
of the graph 𝑦 = 𝑥 by a factor of 2.
2
-5
5
10
GOT IT #3 P.510
8
-2
6
-4
Domain:  x : x  2
Domain:  x : x  4
Asymptotes
x2 xh
Horizontal: y  3
yk
Vertical:
Range:  y : y  3
4
Range:  y : y  6
f x  =
1
x-4
+6
2
5
WRITING THE EQUATION OF A TRANSFORMATION
1
This graph of a function is a translation of the graph of 𝑦 = . What
𝑥
is an equation for the function?
1. Identify the asymptotes.
x  1
y2
4
2. Use the general form.
y
a
k
xh
Homework: p. 512 #19-25 odd, 26-29, 33, 35, 39, 41,
42, 54, 56, 57-63 odd.
2
3. Substitute for a, h and k.
y
1
2
x   1
4. Simplify
y
-2
1
2
x 1
-4
Got It? #4 p.510
y
2
4
x 1
A mystery society is renting a Victorian mansion for a murder mystery party.
The owner is charging the group which has 75 members, $650 for the
weekend rental but does not want more than 30 people in the house. All of
the members attending the party will split the cost of the rental equally
except that the member playing the dead body does not have to pay. Model
the cost per member C as a function of the number of members attending
n. How many members have to attend for the cost to be less than $25 per
person?
The number of members attending: 𝑛
(Remember to subtract the member playing the body)
The number of the paying members: 𝑛 − 1
The cost is divided equally among the paying members
The function is 𝐶 =
650
𝑛−1
Cost must be less than $25 so:
650
n 1
25  n  1  650
25 
650
25
650
n
1
25
n  26
n 1 
650
𝑛−1
In order for the cost to be less than $25, the number of members
attending must be between 28 and 30 inclusive.
2
Homework: p. 512 #.
3