9.1 – inverse variation. variation – direct direct variation is… a relation or function that...
TRANSCRIPT
VARIATION – Direct
• Direct Variation is…
• A relation or function that can be represented by y =kx where k is a constant.
• For example:
• This is a direct variation because the model can be represented by y = 3x
DAYS X = 1 X = 2 X = 3
SNOW LEVEL(Y) 3 6 9
VARIATION – Direct
EX – Y varies directly with X. Y = 100 when x =5.Find k. Then find x when y = 150
VARIATION – Direct
EX – Y varies directly with X. Y = 100 when x =5.Find k. Then find x when y = 150
INVERSE VARIATION
• Inverse Variation is…• A relation or function that can be represented
by xy = k where k is a constant; or – y = k/x
• For example:
• This is an inverse variation because the model can be represented by y = 15/x
DAYS X = 1 X = 2 X = 3
STORE PROFIT(Y) 15 7.5 5
Identifying Direct and Inverse Variation
X 0.5 2 6
y 1.5 6 18
X 2 4 6
y 3.2 1.6 1.1
Ask: What is going on?
y increases as x increases. Since y is three times as big as x each time, this is direct variation; y = 3x
Ask: What is going on?
y increases as x decreases. However, when you multiply x by y you get different values:
6.61.1 6 6.4;1.64 ; 4.62.32
Since there is no clear relationship, we say that there is no variation!
X 0.8 0.6 0.4
y 0.9 1.2 1.8
Ask: What is going on?
y increases as x decreases. Test: is it an inverse variation? What is k?
72.01.8 0.4 72;.01.20.6 ; 72.09.08.0
We have inverse variation.
Therefore, the equation is y = 0.72
x
Joint variation
• Joint variation is variation with more than 2 variables (more than x and y)
• EX - - y varies directly with x and inversely with z.
• This would be y= kx
zThis is the Direct Variation Part
This is the Inverse Variation Part
Joint variation• EX – Y varies directly with x and inversely with z.• Y = 100 when x = 5 and z = 4.• Find k. Then find x when y = 200 and z = 10
Joint variation• EX – Y varies directly with x and inversely with z.• Y = 100 when x = 5 and z = 4.• Find k. Then find x when y = 200 and z = 10
2kxy
3x
ky
kxyz
w
kxyz
wy
kxz
More examples: translate the following:
Y varies directly with the square of x
Y varies inversely with the cube of x
Z varies jointly with x and y
Z varies jointly with x and y and inversely with w
Z varies directly with x and inversely with the product of w and y
Application
• Application – heart rates and life spans of mammals are inversely related.
• Let h = heart rate (bpm) and s = life span (min). The constant, k, is 1,000,000,000.
• That means that hs=1,000,000,000
• Let’s find out your heart rate
Application: Working in groups, find the approximate lifespan of each mammal
MAMMAL HEART RATEBpm
LIFE SPANIn minutes
Mouse 634 1,576,800 = 3 years
Rabbit 158 6,307,200= 12 years
Lion 76 13, 140,000= 25 years
Horse 63 15,768,000= 30 years
Reminder: heart rates and life spans of mammals are inversely related and k = 1,000,000,000
Graphing Inverse Variation
y = k
x
Graphs of inverse functions will look something like the cross between a linear graph and a parabolic curve. In this case, we are just looking at the graph in the first quadrant.
When we look at a true inverse variation function, there will always be two graphs to the functions, diagonal from each other. We will look more at these graphs later on in this chapter.
Graphing Inverse Functions with a Graphing Calculator
1. Press MODE. Scroll down and highlight the word DOT.
2. Press Y= and enter the function 12/x.
Graphing Inverse Functions with a Graphing Calculator
Let’s try x
xf12
)(