Post on 08-Jan-2016
description
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
(For help, go to Lesson 5-5.)
Suppose y varies directly with x. Find each constant of variation.
1. y = 5x 2. y = –7x 3. 3y = x 4. 0.25y = x
Write an equation of the direct variation that includes the given point.
5. (2, 4) 6. (3, 1.5) 7. (–4, 1) 8. (–5, –2)
8-10
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
1. y = 5x; constant of variation = 5
2. y = –7x; constant of variation = –7
3. 3y = x
• 3y = • x
y = x; constant of variation =
4. 0.25y = x
y = x
4 • y = 4 • x
y = 4x; constant of variation = 4
13
1313
13
1414
Solutions
8-10
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Solutions (continued)
5. Point (2, 4) in y = kx: 4 = k(2), so k = 2 and y = 2x.
6. Point (3, 1.5) in y = kx: 1.5 = k(3), so k = 0.5 and y = 0.5x.
7. Point (–4, 1) in y = kx: 1 = k(–4), so k = – and y = – x.
8. Point (–5, –2) in y = kx: –2 = k(–5), so k = and y = x.
14
14
25
25
8-10
ALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Inverse VariationSuppose y varies inversely with x, and y = 9 when x = 8.
Write an equation for the inverse variation.
xy = k Use the general form for an inverse variation.
(8)(9) = k Substitute 8 for x and 9 for y.
72 = k Multiply to solve for k.
xy = 72 Write an equation. Substitute 72 for k in xy = k.
The equation of the inverse variation is xy = 72 or y = .72x
8-10
ALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Inverse VariationThe points (5, 6) and (3, y) are two points on the graph of an
inverse variation. Find the missing value.
x1 • y1 = x2 • y2 Use the equation x1 • y1 = x2 • y2 since you know coordinates, but not the constant of variation.
5(6) = 3y2 Substitute 5 for x1, 6 for y1, and 3 for x2.
30 = 3y2 Simplify.
10 = y2 Solve for y2.
The missing value is 10. The point (3, 10) is on the graph of the inverse variation that includes the point (5, 6).
8-10
ALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Inverse VariationJeff weighs 130 pounds and is 5 ft from the lever’s fulcrum. If
Tracy weighs 93 pounds, how far from the fulcrum should she sit in
order to balance the lever?
Relate: A weight of 130 lb is 5 ft from the fulcrum.A weight of 93 lb is x ft from the fulcrum.Weight and distance vary inversely.
Define: Let weight1 = 130 lbLet weight2 = 93 lbLet distance1 = 5 ftLet distance2 = x ft
8-10
ALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Inverse Variation(continued)
650 = 93x Simplify.
Tracy should sit 6.99, or 7 ft, from the fulcrum to balance the lever.
Write: weight1 • distance1 = weight2 • distance2
130 • 5 = 93 • x Substitute.
6.99 = x Simplify.
= x Solve for x.65093
8-10
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Decide if each data set represents a direct variation or an
inverse variation. Then write an equation to model the data.
x y
3 10
5 6
10 3
a. The values of y seem to vary inversely with the values of x.
Check each product xy.
xy: 3(10) = 30 5(6) = 30 10(3) = 30
The product of xy is the same for all pairs of data.
So, this is an inverse variation, and k = 30.
The equation is xy = 30.
8-10
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
(continued)
x y
2 3
4 6
8 12
b. The values of y seem to vary directly with the values of x.
So, this is a direct variation, and k = 1.5.
The equation is y = 1.5x.
64 = 1.5 = 1.5
128
yx = 1.5
32
The ratio is the same for all pairs of data.yx
Check each ratio .yx
8-10
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
Explain whether each situation represents a direct variation
or an inverse variation.
b. The cost of a $25 birthday present is split among several friends.
Since the total cost is a constant product of $25, this is an inverse variation.
The cost per souvenir times the number of souvenirs equals the total cost of the souvenirs.
Since the ratio is constant at $10 each, this is a direct variation. cost souvenirs
8-10
The cost per person times the number of people equals the total cost of the gift.
a. You buy several souvenirs for $10 each.
Inverse VariationALGEBRA 1 LESSON 8-10ALGEBRA 1 LESSON 8-10
1. The points (5, 1) and (10, y) are on the graph of an inverse variation. Find y.
2. Find the constant of variation k for the inverse variation where a = 2.5 when b = 7.
4. Tell whether each situation represents a direct variation or an inverse variation.a. You buy several notebooks for $3 each.
b. The $45 cost of a dinner at a restaurant is split among several people.
3. Write an equation to model the data and complete the table.
0.5
17.5
direct variation
Inverse variation
x y
1
2
6
131619
xy =13 3
1 18
8-10