About Graph of a Proportional Relationships
What are Proportional relationships?
Definition: When two quantities are compared simultaneously, they are said to be in a ratio. The quantities in the ratio are expressed in the form of \(\frac{a}{b}\). The ratio of a to b can also be written as a:b. In a manner of speaking, if five cupcakes can be made from one package of cupcake mix, then, this is the same as saying 10 cupcakes can be made from two packages of cake mix.
Two quantities, x and y are said to be proportional when y = kx, where k is a number and \(k\neq 0\). The “k” is called the constant of proportionality.
Determining Proportional relationships by graphs
The graph of y = kx is shown here. It is a line that passes through the origin. In the example given below, two sets of values are given and the graph is plotted. After plotting the graph, the dots are connected. If the line passes through the origin it is said to be proportional.
In the first graph, the values are not proportional. The second set of values are proportional as the line passes through the origin.
How to find unit rates from graphs?
We take any point on the graph (x,y). There are two ways to find the unit rate.
- When the x-coordinate is taken as 1: For this we divide both the coordinates (x,y) by x, which results as \((1, ~\frac{y}{x})\). So every 1 unit on the x-axis corresponds to \(\frac{y}{x}\) units on the y axis.
- When the y-coordinate is taken as 1: For this we divide both the coordinates (x,y) by y, which results as \(( \frac{x}{y},~1)\). So every 1 unit on the y axis corresponds to \(\frac{x}{y}\) units on the x axis.
Solved Graphs of Proportional Relationship Examples
Example 1:
Plot the points on a graph paper and check if they represent a proportional relationship.
Hours, x | Miles, y |
|---|---|
0 | 1 |
2 | 2 |
4 | 3 |
The points do not pass through the origin, hence the graph does not have a proportional relationship.
Example 2:
Apples, x | Grapes, y |
|---|---|
4 | 8 |
6 | 12 |
8 | 16 |
The points when plotted and extended backwards meet at the origin. This means that a proportional relationship exists.
Example 3:
x and y are proportional in both the scenarios, use the values to find the constant of proportionality.
1. When y = 74, x = 8,
Solution:
y = kx
y =74, x = 8
\(k=\frac{74}{8}=\frac{37}{4}\)
2. When y = 24, x = 12
y = kx
y = 24, x = 12
\(k=\frac{24}{12}=2\)
Example 5:
Given below is a table that shows profit earned by selling copper wires. Find the profit earned for selling 80 pounds of copper
Copper wire (lb), x | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
Profit, y | 5 | 10 | 15 | 20 |
Answer:
When the points are plotted on a graph, the points form a straight line and they pass the origin as well. This means that the graph is proportional. The unit rate in this situation can be calculated in the following way:
Let’s take a point (10,5) on the graph, on dividing by 10 we get it as (1,0.5), that is for every 1 pound of copper wire, the profit is 0.5 dollars. Using the graph also, we get this point on the graph (1,0.5).
So for 80 pounds of copper wire, the profits are as follows
\(0.5\times80=40\) dollars
Therefore the profits earned from selling 80 pounds of copper wire is 40 dollars.
A unit rate is when one quantity is compared to only 1 unit of another quantity. For example, covering a distance of 2 meters every second, climbing 1 step every 5 seconds, etc.
If a line has to be a proportional relationship, then the line has to pass through the origin. If the line has any intercepts, then that means the line does not pass through the origin. Hence, if the graph has intercepts, then it cannot be a proportional relationship.