2.6 Inverse Variation Graphs: 𝒚 =
𝒌
AND 𝒚 =
𝒙
𝒌
𝒙𝟐
SKILLS TO ACQUIRE:
 RECOGNIZE PROPERTIES OF THE GRAPHS
 GRAPH VARIATION EQUATION
VOCABULARY
 Branches: Parts of a graph (not linear!)
 Vertical Asymptote (VA): An imaginary vertical line that the graph gets close to but does not
cross
 Horizontal Asymptote (HA): An imaginary horizontal line that the graph gets close to but
does not cross
Inverse Variation Graph:
GRAPH:
1
𝟒
𝑦 = 𝑥 𝑎𝑛𝑑 𝒚 = 𝒙
𝒚=
𝒌
𝑿
GRAPH:
1
𝟏
𝑦 = 𝑥 𝑎𝑛𝑑 𝒚 = 𝟒𝒙
Type of Graph: Hyperbola
GRAPH:
1
−𝟏
𝑦 = 𝑥 𝑎𝑛𝑑 𝒚 = 𝒙
K>0:
DOMAIN: x cannot be 0
LINE OF SYMMETRY: y=x or y=-x
VA: y-axis
RANGE: y cannot be 0
HA: x-axis
QUADRANTS: 1 and 3
K<0:
DOMAIN: x cannot be 0
Note: BRANCHES DIAGONAL
0 < K < 1 or -1 < K < 0: graph closer to origin
K > 1: graph farther from origin
Inverse Variation Graph: 𝒚 =
RANGE: y cannot be 0
QUADRANTS: 2 and 4
𝒌
𝑿𝟐
GRAPH:
1
2
𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑦 = 𝑥 2
GRAPH:
1
1
𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑦 = 2𝑥 2
Type of Graph: Inverse Square curve
LINE OF SYMMETRY: y-axis
GRAPH:
1
−1
𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑦 = 𝑥 2
K>0:
DOMAIN: x does not equal 0
RANGE: y > 0
VA: y-axis
QUADRANTS: 1 and 2
HA: x-axis
K<0:
DOMAIN: x does not equal zero
Note: TREE TRUNK BRANCHES
0 < K < 1 or -1 < K < 0: curves closer to y-axis
RANGE: y < 0
K > 1: curves farther away from y-axis
QUADRANTS: 3 and 4
Example: Given the equation 𝒚 =
−𝟒
𝒙𝟐
(a) Find the rate of change between x = -2 and x = 1 m = -1
(b) Find the rate of change between x = 2 and x = 4 m = 3/8
Go For It!
Which of the four graphs match the following descriptions?
There can be more than one answer!
a) y = kx
b) y = kx2
c) y=k/x
d) y=k/x2
1.) Two lines of symmetry?
2.) Asymptotes?
3.) When K < 0, have graphs in the first quadrant?
4.) Is a hyperbola?
5.) Constant slope?
6.) Has a domain of all real numbers?
7.) Has a range of all real numbers except 0?
8.) Has its branches side by side?
9.) When K > 0, graph is in the 2nd quadrant?
10.) When K < 0, has a range y < 0.
𝟓
11.) Sketch the graph of 𝒚 = 𝒙
𝟏
12.) Sketch a graph of 𝒚 = 𝟓𝒙𝟐