aa section 2-1
TRANSCRIPT
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Section 2-1Direct Variation
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r Varies Directly as c:
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation:
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
Direct Variation Function:
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0
Can also be known as “directly proportional”
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0
Can also be known as “directly proportional”***The cost of gas varies directly as the amount of
gas purchased
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r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c
Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer
Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0
Can also be known as “directly proportional”***The cost of gas varies directly as the amount of
gas purchasedThe more you get, the more it costs
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Example 1
Rewrite the statement, “The cost of gas varies directly as the amount of gas purchased.”
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Example 1
Rewrite the statement, “The cost of gas varies directly as the amount of gas purchased.”
“The cost of gas is directly proportional to the amount of gas purchased.”
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.P = kE
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.P = kE
b. Identify the independent and dependent variables.
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.P = kE
b. Identify the independent and dependent variables.
Independent: E
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.P = kE
b. Identify the independent and dependent variables.
Independent: E Dependent = P
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Example 2
The weight of an object on planet P varies directly with its weight on Earth E.
a. Write an equation relating P and E.P = kE
b. Identify the independent and dependent variables.
Independent: E Dependent = Pk is just a constant
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Example 3
The ingredients for a pizza and the price are proportional to its area. This means the quantity of
ingredients is proportional to the square of its radius. Suppose a pizza 12 in. in diameter costs $7.00. If the price varies directly as the square of its radius, what would a pizza 16 in. in diameter cost? What
about an 18 in. pizza?
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Example 3
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Example 3
c = cost
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Example 3
c = cost r = radius
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Example 3
c = cost r = radiusc = kr2
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36)
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64)
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64) ≈ $12.44
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64) ≈ $12.44
c = 736 (9)
2
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64) ≈ $12.44
c = 736 (9)
2 = 736 (81)
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64) ≈ $12.44
c = 736 (9)
2 = 736 (81) = $15.75
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Example 3
c = cost r = radiusc = kr2
7 = k(6)2 7 = k(36) k = 736 c = 7
36 r2
c = 736 (8)
2 = 736 (64) ≈ $12.44
c = 736 (9)
2 = 736 (81) = $15.75
A 16 in. diameter pizza would cost $12.44 and an 18 in. diameter pizza would cost $15.75.
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Steps to solving a direct variation problem
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Steps to solving a direct variation problem
1.Write an equation to describe the variation
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Steps to solving a direct variation problem
1.Write an equation to describe the variation
2.Find k
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Steps to solving a direct variation problem
1.Write an equation to describe the variation
2.Find k3.Rewrite the function using k
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Steps to solving a direct variation problem
1.Write an equation to describe the variation
2.Find k3.Rewrite the function using k4.Evaluate
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
32 = k(2)
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
32 = k(2)
k = 16
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
32 = k(2)
k = 16
y = 16x
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
32 = k(2)
k = 16
y = 16x
y = 16(5)
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Example 4
Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.
y = kx
32 = k(2)
k = 16
y = 16x
y = 16(5) = 80
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn48 = k(12)
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn48 = k(12)
k = 4
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn48 = k(12)
k = 4m = 4n
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn48 = k(12)
k = 4m = 4n
m = 4(3)
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Example 5
m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.
m = kn48 = k(12)
k = 4m = 4n
m = 4(3) = 12
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Homework
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Homework
p. 74 #1 - 25