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Warm-Up
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Sections 2.1 and 2.2 Direct and Inverse Variation
Chapter 2: Variations
Essential Question:
• What are the differences between direct and inverse variation?
Direct Variation
Direct Variationwhere k is a nonzero constant
and n is a positive number
€
y = kxn
Direct Variation
We say this “y is directly proportional to x”
where k is a nonzero constant and n is a positive number
€
y = kxn
Direct Variation
We say this “y is directly proportional to x”
When one variable increases then the other variable increases
where k is a nonzero constant and n is a positive number
€
y = kxn
Direct Variation
We say this “y is directly proportional to x”
When one variable increases then the other variable increases
also the opposite - one decreases the other decreases
where k is a nonzero constant and n is a positive number
€
y = kxn
Examples:1. The cost of gas for a car varies directly as the amount of gas purchased.
C = costA = amount
k depends on the economy
Equation:
Examples:1. The cost of gas for a car varies directly as the amount of gas purchased.
C = costA = amount
k depends on the economy
Equation:
C = kA
Examples:1. The cost of gas for a car varies directly as the amount of gas purchased.
C = costA = amount
k depends on the economy
Equation:
C = kA
2. The volume of a sphere varies directly as the cube of its radius.
Examples:1. The cost of gas for a car varies directly as the amount of gas purchased.
C = costA = amount
k depends on the economy
Equation:
C = kA
2. The volume of a sphere varies directly as the cube of its radius.
Equation:
Examples:1. The cost of gas for a car varies directly as the amount of gas purchased.
C = costA = amount
k depends on the economy
Equation:
C = kA
2. The volume of a sphere varies directly as the cube of its radius.
Equation:
V = kr3
Inverse Variation
Inverse Variationwhere and n > 0.
€
y =kxn
€
k ≠ 0
Inverse Variationwhere and n > 0.
€
y =kxn
€
k ≠ 0
We say “y is inversely proportional to x”
Inverse Variationwhere and n > 0.
€
y =kxn
€
k ≠ 0
We say “y is inversely proportional to x”
When one variable increases then the other variable decreases or vice versa
Examples
Examples3. m varies inversely with n2
Examples3. m varies inversely with n2
€
m =kn2
Examples3. m varies inversely with n2
€
m =kn2
4. The weight W of a body varies inversely with the square of its distance d from the center of the earth.
Examples3. m varies inversely with n2
€
m =kn2
4. The weight W of a body varies inversely with the square of its distance d from the center of the earth.
€
W =kd2
Four Steps to Predict the Values of Variation Functions:
Four Steps to Predict the Values of Variation Functions:
1. Write an equation that describes the variation
Four Steps to Predict the Values of Variation Functions:
1. Write an equation that describes the variation
2. Find the constant of variation (k)
Four Steps to Predict the Values of Variation Functions:
1. Write an equation that describes the variation
2. Find the constant of variation (k)
3. Rewrite the variation function using k.
Four Steps to Predict the Values of Variation Functions:
1. Write an equation that describes the variation
2. Find the constant of variation (k)
3. Rewrite the variation function using k.
4. Evaluate the function for the desired value of the independent variable.
Examples:
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3)
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.)
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
k = 40
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
k = 40
(3.)
€
y =4063
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
k = 40
(3.)
€
y =4063
(4.)
€
y =40216
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
k = 40
(3.)
€
y =4063
(4.)
€
y =40216
€
y =527
Examples:5. m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
(1.) m = kn (2.) 48 = k(12)
k = 4
(3.) m = 4(3) (4.) m = 12
6. y varies inversely as the cube of x. If y = 5 when x = 2, find y when x = 6.
€
y =kx 3
(1.) (2.)
€
5 =k23
€
5 =k8
k = 40
(3.)
€
y =4063
(4.)
€
y =40216
€
y =527
Last ONE!
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
k = 7
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
k = 7
(3.)
€
y = 7(9)2
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
k = 7
(3.)
€
y = 7(9)2 (4.)
€
y = 7(81)
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
k = 7
(3.)
€
y = 7(9)2 (4.)
€
y = 7(81)
y = 567
Last ONE!7. y varies directly as the square of x. If y = 63 and x = 3, find y when x = 9.
(1.)
€
y = kx 2 (2.)
€
63 = k32
€
63 = 9k
k = 7
(3.)
€
y = 7(9)2 (4.)
€
y = 7(81)
y = 567
Summarizer:
Summarizer:1. What is the formula for inverse variation?
Summarizer:1. What is the formula for inverse variation?
where and n > 0.
€
y =kxn
€
k ≠ 0
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
where and n > 0.
€
y =kxn
€
k ≠ 0
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?up
where and n > 0.
€
y =kxn
€
k ≠ 0
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
€
y = kxn
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
4. For V = :
€
43πr2
up
where and n > 0.
€
y =kxn
€
k ≠ 0
€
y = kxn
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
4. For V = :
€
43πr2
a. What is the constant of variation?
b. What is the independent variable?
c. What is the dependent variable?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
€
y = kxn
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
4. For V = :
€
43πr2
a. What is the constant of variation?
b. What is the independent variable?
c. What is the dependent variable?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
€
43π
€
y = kxn
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
4. For V = :
€
43πr2
a. What is the constant of variation?
b. What is the independent variable?
c. What is the dependent variable?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
r€
43π
€
y = kxn
Summarizer:1. What is the formula for inverse variation?
2. For inverse, when one variable goes down the other variable goes _________?
3. What is the formula for direct variation?
4. For V = :
€
43πr2
a. What is the constant of variation?
b. What is the independent variable?
c. What is the dependent variable?
up
where and n > 0.
€
y =kxn
€
k ≠ 0
rV€
43π
€
y = kxn
Homework:
2.1 A Worksheetand
2.2 A Worksheet
