5.05 Proportional relationships in the coordinate plane

Proportional relationships can be represented in a table, in an equation, verbally, or graphically.

Before we discuss graphing proportional relationships, we will review the coordinate plane:

• A number plane is created by two perpendicular lines that we call an $x$x-axis and a $y$y-axis.  
• The $x$x-axis is the horizontal line and the $y$y-axis is the vertical line.
• Where the two axes cross each other is labeled the origin.  It has a zero value on both axes.
• Both axes have positive and negative values which are divided by the origin as shown in the diagram below.

We can create a grid from the two number lines. When labeling points on the grid, we always use the $x$x-value first.  

Representing a ratio graphically

We know that a ratio compares the relationship between two values; it compares how much there is of one thing compared to another. We can therefore plot pairs of ratios on a number plane.

Representing a ratio as an equation

Proportional relationships can also be written as linear equations.

We can solve proportional relationships like equations to make judgments about the relationship between two variables. We can also graph these equations. 

The diagram below shows a graph where Variable $2$2 is directly proportional to Variable $1$1:

Every proportional equation has a graph similar to this one. It is a straight line, and goes through the point $\left(0,0\right)$(0,0). This is because if $x=0$x=0 then $y$y must be equal to $0$0 as well. The constant of proportionality is what determines how steep the line is.

Worked examples

Question 2

Joe serves $2$2 hot dogs every $4$4 minutes. 

Using $y$y for the number of hot dogs and $x$x for the amount of minutes that have passed, write an equation that represents this proportional relationship.

Think: The equation for a proportional relationship is $y=kx$y=kx. We can substitute two of these variables to solve for the constant of proportionality, $k$k.

Do:

Now, we substitute the constant of proportionality back into the general equation:

Reflect: How long would it take Joe to serve $12$12 hot dogs?

Question 3

The price for playing arcade games is shown in the graph with price vs number of games.

What does the point $\left(3,5\right)$(3,5) represent on the graph?

Think: Which variable does each axis represent?

Do:

The horizontal axis tells us the price. Since the $x$x-coordinate corresponds to the horizontal axis, this means the price is $\$3$$3.

Similarly, the vertical axis tells us the number of games played. The $y$y-coordinate corresponds to the vertical axis, so this means $5$5 games were played.

Thus, $5$5 games cost $\$3$$3.

Reflect: How many games could you play for $\$12$$12?

Practice questions

Question 4

Question 5

Question 6

Units rates from a graph

Recall that a rate is a special type of ratio that is used to compare different types of quantities.

A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

Some common unit rates are distance per hour, cost per item, earnings per week, etc. Do you see how in each example the first quantity is related to one unit of the second quantity?

When we are looking at unit rates on graphs, we want to know how much the dependent ($y$y) variable will increase by when the independent ($x$x) variable is increased by one.

Practice questions

Question 7

Question 8