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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{)}010203040
\text{Miles } (y\text{)}015304560

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 1.5Evaluate 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:

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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 1.5Evaluate the division

Examples

Example 1

The table below shows the cost of various amounts of gas in dollars:

\text{Number of gallons } (x\text{)}010203040
\text{Cost of gas } (y\text{)}044.0088.00132.00176.00
a

Find the unit rate in \$ per gallons.

Worked Solution
Create a strategy

Find the unit rate by getting the constant of proportionality from the table of values. Divide the cost of gas in dollars by the number of gallons.

Apply the idea
\displaystyle \text{ Unit rate}\displaystyle =\displaystyle \dfrac{44}{10} Choose any ordered pair.
\displaystyle =\displaystyle 4.4 Evaluate the division

From the table, the constant of proportionality which is also the unit rate, is \$4.4/\text{gal}

b

Graph the proportional relationship based on the unit rate.

Worked Solution
Create a strategy

The unit rate is also the slope of the line representing the proportional relationship. The slope is the rise over run.

Apply the idea

The unit rate 4.4 is also the slope which means a rise of 4.4 from the origin and a run of 1 to the right.

We also know that graphs of proportional relationships always go through the origin, so our graph will start at \left(0,0\right).

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Reflect and check

We can also use the values of the table to plot the graph of the relationship.

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Idea summary

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.

Outcomes

8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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