4.02 Proportional relationships in the coordinate plane
Introduction
We have learned that slope and unit rate and the constant of proportionality, k, are all equal. In this lesson, we will use the value of the slope to graph proportional relationships in the coordinate plane.
Proportional relationships in the coordinate plane
The constant of proportionality, unit rate, or slope can be computed in various ways and are helpful when looking at proportional relationships in the coordinate plane. They can be identified by looking at coordinates in a table of values, from an equation of a line, or from coordinates on a graph.
For example, consider the following proportional relationship represented with a table of a values:
| \text{Minutes } (x\text{)} | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Miles } (y\text{)} | 0 | 15 | 30 | 45 | 60 |
We can calculate the unit rate of the relationship by choosing one of the ordered pairs and dividing. Note that we won't choose the point \left(0,0\right) because we can't divide by 0:
| \displaystyle \text{Unit rate} | \displaystyle = | \displaystyle \dfrac{15}{10} | Choose any ordered pair |
| \displaystyle \text{Unit rate} | \displaystyle = | \displaystyle 1.5 | Evaluate the division |
Therefore the unit rate of the proportional relationship is \dfrac{1.5 \text{ miles}}{\text{minutes}}.
We can look at the same proportional relationship as an equation: y=1.5x
The constant of proportionality is identified as the coefficient of x. Therefore 1.5 is the constant of proportionality.
Lastly, we can represent this proportional relationship in a coordinate plane:
We can identify the slope of the proportional relationship by calculating the rise and dividing it by the run:
| \displaystyle \text{Slope} | \displaystyle = | \displaystyle \dfrac{\text{rise}}{\text{run}} | Formula for slope |
| \displaystyle \text{Slope} | \displaystyle = | \displaystyle \dfrac{15}{10} | Determine the rise and run of the line |
| \displaystyle \text{Slope} | \displaystyle = | \displaystyle 1.5 | Evaluate the division |
Examples
Example 1
The table below shows the cost of various amounts of gas in dollars:
| \text{Number of gallons } (x\text{)} | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of gas } (y\text{)} | 0 | 44.00 | 88.00 | 132.00 | 176.00 |
Find the unit rate in \$ per gallons.
Graph the proportional relationship based on the unit rate.
Every proportional equation has a graph of a straight line that goes through the point (0, 0). This is because if x = 0 then y must be equal to 0 as well. The constant of proportionality is what determines how steep the line is.
A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.
We can graph a proportional relationship given the unit rate, constant of proportionality or slope. These determine how much the dependent (y) variable will increase by when the independent (x) variable is increased by one.