Section 2-3The Fundamental Theorem of Variation

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area:

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8)

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side:

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5)

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

New area:

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

New area: 12(12)

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

New area: 12(12) = 144 ft2

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

New area: 12(12) = 144 ft2

14464

Warm-up1. Each side of a square patio is 8 feet long. Matt

Mitarnowski plans to increase the size of the patio by 50%. How will the area of the enlarged patio compare

with the area of the original?Original area: 8(8) = 64 ft2 Side = 8 ft

New side: 8(1.5) = 12 ft

New area: 12(12) = 144 ft2

14464

= 2.25 times larger

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube:

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length:

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3)

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

New volume:

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

New volume: 273

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

New volume: 273 = 19683 in3

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

New volume: 273 = 19683 in3

19683729

Warm-up2. The length of each edge of a cube is 9 in. How does

the volume of a cube with edges 3 times as long compare with the volume of the smaller cube?

Volume of smaller cube: 93 = 729 in3

Enlarged edge length: 9(3) = 27 in

New volume: 273 = 19683 in3

19683729

= 27 times larger

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

b. Now complete a second table, doubling the radii in the above table.

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

b. Now complete a second table, doubling the radii in the above table.

r 2 4 6 8 10 12 14

A 4π 16π 36π 64π 100π 144π 196π

Example 1a. Complete the following table in terms of π.

r 1 2 3 4 5 6 7

A π 4π 9π 16π 25π 36π 49π

b. Now complete a second table, doubling the radii in the above table.

r 2 4 6 8 10 12 14

A 4π 16π 36π 64π 100π 144π 196π

Example 1c. How do the areas of the new table compare with the

areas of the first? Do you notice a pattern?

Example 1c. How do the areas of the new table compare with the

areas of the first? Do you notice a pattern?

When the radius is doubled, the area is quadrupled

Fundamental Theorem of Variation

Fundamental Theorem of Variation

a. If y varies directly as xn and x is multiplied by c, then y is multiplied by cn.

Fundamental Theorem of Variation

a. If y varies directly as xn and x is multiplied by c, then y is multiplied by cn.

b. If y varies inversely as xn and x is multiplied by c when c ≠ 0, then y is divided by cn.

Fundamental Theorem of Variation

a. If y varies directly as xn and x is multiplied by c, then y is multiplied by cn.

b. If y varies inversely as xn and x is multiplied by c when c ≠ 0, then y is divided by cn.

Proof?

Example 2

tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the

intensity of the light?

I =

k

D2

Example 2

tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the

intensity of the light?

I =

k

D2

I =

k

D2

Example 2

tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the

intensity of the light?

I =

k

D2

I =

k

D2 (2D)2

Example 2

tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the

intensity of the light?

I =

k

D2

I =

k

D2 (2D)2 = 4D2

Example 2

tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the

intensity of the light?

I =

k

D2

I =

k

D2 (2D)2 = 4D2

We are now dividing k by 4D2, so I is also divided by 4

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

y

2=

k

2x

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

y

2=

k

2x

y is divided by 2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

y

2=

k

2x

y is divided by 2 y =

k

x2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

y

2=

k

2x

y is divided by 2 y =

k

x2

y

4=

k

(2x )2

Example 3Suppose that in a variation problem, the value of x is

doubled. What is the effect on y if:

a. y varies directly as x b. y varies directly as x2

c. y varies inversely as x d. y varies inversely as x2

y = kx2y = k(2x)

y is doubled

y = kx2

4y = k(2x)2

y is multiplied by 4 (22)

y =

k

x

y

2=

k

2x

y is divided by 2 y =

k

x2

y

4=

k

(2x )2

y is divided by 4 (22)

Homework

Homework

p. 87 #1 - 23