Representing Proportional Relationships
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Representing Proportional Relationships
A directly proportional relationship has an equation in the form of y=kx. It is a relationship between two quantities in
which one is a constant multiple of the other. When one quantity changes, the other changes by a constant factor, k. The constant factor k is the constant of proportionality.
Example: Christopher cuts grass (any size yard) for $10.00.
How much money would he make if he cut 3 yards? 5 yards?
0 yards?
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The function table below shows the relationship between the side lengths of a regular octagon and its perimeter.
Step 1: Write an equation to represent the situation. (the perimeter is always 8 times the side length of a regular octagon. So, 8 is
the constant of proportionality)
P = 8sStep 2: Substitute the side length of 9 for s and find the perimeter. P = 8 x 9
P = 72 in.
Side lengths s (inches)
Perimeter, P (inches)
1 8
2 16
3 24
4 32
5 ?
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A directly proportional relationship is a linear relationship because it forms a straight line when graphed. The graph of a proportional relationship is a straight line that passes through the origin (0,0). It will also pass through the point (1,k) where k is the constant of proportionality, or the unit rate.
Example: An empty swimming pool is being filled at a rate of 10 gallons per minute. Make a graph to display the amount of water in the pool each minute for 6 minutes.
Step 1: Write an equation that represents the situation. Let x = the number of minutes and y = the number of gallons y = 10x
Step 2: Make a table to show the number of gallons in the pool each minute.# of min
(x)# of gal(y)
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# of min (x)
0 1 2 3 4 5 6
# of gal(y)
0 10 20 30 40 50 60
Step 3: Make a line graph, using the ordered pairs in the table.
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The graph below shows the amounts charged for purchasing different number roses from a florist. Is there a proportional relationship between the number of roses bout and the cost? If so, what is the constant of proportionality and what does it mean in context?
Step 1: Think about the graph of a proportional relationship.
Step 2: Determine the constant of proportionality.
Step 3: Determine what the constant of proportionality means in context.
The relationship is proportional and the constant of proportionality, 2, means that each rose costs $2.
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Check it out!!The table shows the relationship between the side lengths of a regular pentagon and its perimeter. Which equation shows the relationship between the side length and the perimeter of a regular pentagon.
A.P = s + 5B.P = 5sC.P = 1/5sD.P = 5s + 5
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If the regular pentagon (in the last problem) has side lengths of 8 inches, what is its perimeter?
A. 13 inchesB. 30 inchesC. 40 inchesD. 45 inches
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The graph shows the relationship between the cost of the number of uniforms ordered by a sports team. Which equation shows the relationship between the number of uniforms, x, and the cost, y?
A. Y = 20xB. Y = 10xC. Y = 2xD. Y = x
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What is the unit cost of a uniform in the last problem?
A. $2 per uniformB. $5 per uniformC. $10 per uniformD. $20 per uniform
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Use the information in the last problem to determine how much it will cost the team to order 8 uniforms?
A. $20B. $140C. $160D. $180
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