We've already learned about proportional relationships, where two quantities vary in such a way that one is a constant multiple of the other. In other words, they always vary by the same constant.
Proportional relationships are a special kind of linear relationship that can be written generally in the form $y=kx$y=kx and always pass through the origin $\left(0,0\right)$(0,0).
To interpret information from a graph, we need to look at pairs of coordinates. Coordinates tell us how one variable relates to the other. Each pair has an $x$x value and a $y$y value in the form $\left(x,y\right)$(x,y).
It doesn't matter what labels we give our axes, this order is always the same.
Let's look at some examples and see this in action.
The number of eggs farmer Joe's chickens produce each day are shown in the graph.
What does the point $\left(6,3\right)$(6,3) represent on the graph?
Think: The first coordinate corresponds to the values on the $x$x-axis (which in this problem would represent the number of days) and the second coordinate corresponds to values on the $y$y-axis ( which in this problem would represent the number of eggs).
Using the given information in context, we can interpret this point to mean that in $6$6 days the chickens will produce $3$3 eggs.
Reflect: How many days does it take for the chickens to lay $1$1 egg?
The number of liters of gas used by a fighter jet over a certain number of seconds is shown in the graph. What does the point on the graph represent?
The number of cupcakes eaten at a party is shown on the graph.
What does the point on the graph represent?
Danielle is having a party, and expects to have $10$10 guests. According to the rate shown on the graph, how many cupcakes should she buy?
Recognize and represent proportional relationships between quantities.
(d) Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.