Inverse Variation
Objectives: To solve inverse variation
Write an equation given a point.
Find the missing coordinate
To compare inverse variation and direct variation
Solve application problems involving inverse variation.
Suppose you are part of a volunteer crew constructing affordable housing. Building a house requires a total of 160 workdays. For example, a crew of 20 people can complete a house in 8 days.
How long should it take a crew of 40 people?
Complete the table below.
Crew Size (x) |
Construction Days (y) |
Total Workdays |
2 |
80 |
160 |
5 |
|
160 |
8 |
|
|
|
16 |
160 |
20 |
8 |
160 |
40 |
|
|
Graph the (x, y) data in the table above.
Describe what happens to construction time as the crew size increases.
Inverse Variation
A function in the form xy = k or , where k ≠ 0, is an inverse variation. The constant of variation for direct variation k is the product of x and y for an ordered pair (x, y)
Inverse Variations all have graphs with the same general shape.
H ow does k affect the shape of the graph?
Examples
Graph: xy = 2
xy = 6
xy = 12
Describe the general shape of the graph of an inverse variation.
How is it different than a graph with direct variation? (Like y = 2x)
Examples – Suppose y varies inversely with x. Write an equation for the inverse variation.
y = 6 when x = 3 b. y = 7 when x = 8 c. y = 6 when x = 1/3
y = 9 when x = 2 e. y = 7 when x = 5
Suppose (x1,y1) and (x2,y2) are ordered pairs of an inverse variation. Each ordered pair ______________________________________________________________________
Example – Finding the Missing Coordinate
Each pair of points is on the graph of an inverse variation. Find the missing value.
(3,8) and (2,y)
(3,y) and (5,9)
(75, 0.2) and (x, 3)
Example – Real world Problem Solving
The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. Where should Julio, who weighs 150 lb, sit to balance the lever?
A 100-lb weight is placed 4 ft from a fulcrum. How far from the fulcrum should a 75-lb weight be placed to balance the lever?
An 80-lb weight is placed 9 ft from the fulcrum. What weight should you put 6 ft from the fulcrum to balance the lever?
Comparing Direct and Inverse Variation
Direct Variation: y = kx
Inverse Variation: xy = k
G raph the following equations on the graph to the right. Label and Identify as Direct or Inverse Variation. (Make a table of values to graph)
y = 3x
Determine whether the data in each table represent a direct variation or an inverse variation.
How?? Look at the relationship between x and y values. Which equation does it fit:
y = kx or xy = k?
a. b.
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x |
y |
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|
2 |
5 |
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|
4 |
10 |
||||
|
10 |
25 |
||||
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|
x |
y |
5 |
20 |
10 |
10 |
25 |
4 |
c. d.
x |
y |
3 |
12 |
6 |
6 |
9 |
4 |
x |
y |
3 |
12 |
5 |
20 |
8 |
32 |
Examples – Application Problems
Explain whether each situation represents a direct or an inverse variation.
The cost of $20 worth of gasoline is split among several people.
You buy several markers for 70¢ each.
You are in a discount store. All sweaters are on sale for $15 each.
You walk 5 miles each day. Your speed and time vary from day to day.
Experiment 7 – Radioactivity and the Inverse Square law
FUNZIONI INVERSE SIMMETRIE RISPETTO ALLA BISETTRICE DEI QUADRANTI DISPARI
HOMEWORK WORKSHEET 45 SIMPLIFYING RADICALS & SOLVING WITH INVERSE
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