Reading and Drawing Proportional Graphs in Context

Every pair with $x \neq 0$ gives the same ratio, $2$, and the stand charges that same rate for any weight. So $k = 2$ dollars per pound and $y = 2x$. The pair with $x = 0$ fits this rule too, because $2 \cdot 0 = 0$.

The figure below graphs the relationship. The horizontal axis shows the weight $x$ in pounds, and the vertical axis shows the cost $y$ in dollars. Each row of the table becomes one labeled point: $(0, 0)$, $(1, 2)$, $(3, 6)$ and $(4, 8)$. All four pairs satisfy $y = 2x$, so the points lie on one straight line through the origin. That line is Line A.

Now read a point as a sentence. A point is written as a coordinate pair $(x, y)$. The first number is always the horizontal quantity, and the second is always the vertical quantity. So $(3, 6)$ says: 3 pounds of apples cost 6 dollars. One point gives you two matching facts, one from each axis.

Now turn the idea around. Instead of reading a finished graph, you will build one from a word problem.

First decide whether the story describes a proportional relationship. It does when $y/x$ is the same constant $k$ for every pair with $x \neq 0$. A steady rate that starts from zero gives you this: with $k$ units of $y$ for each unit of $x$, the total is $y = kx$.

Then collect exact pairs:

• The story gives one pair, such as 8 kilometers in 2 hours. Divide its $y$ by its $x$, which must not be $0$, to find $k$.
• Because $y = kx$, the pairs $(0, 0)$ and $(1, k)$ belong to the graph too.

Plot your pairs and label them. Because they all satisfy $y = kx$, they lie on one straight line that starts at $(0, 0)$, so you can draw it. Compute every pair from the story. Never guess one from how the picture looks.

The figure above is Dana's graph. The horizontal axis shows time in hours, and the vertical axis shows distance in kilometers. Each labeled point is one exact pair from the example, and Line A passes through all four because each pair satisfies $y = 4x$. You can read any of them as a sentence: $(2, 8)$ says that Dana covers 8 kilometers in 2 hours.

Dana's relationship is proportional because her steady pace starts from zero. The next example tests a steady rate that does not start from zero.

Here is the idea to keep. A point $(x, y)$ on a graph is a pair of matching facts: $x$ units of the horizontal quantity go with $y$ units of the vertical quantity. When $y/x$ is one constant $k$ for every pair with $x \neq 0$, so that $y = kx$ and the graph is a straight line through $(0, 0)$, the point $(0, 0)$ says that zero goes with zero, and the point $(1, k)$ shows the unit rate. To graph a word problem, name the quantities and their units, compute exact pairs from the story, check the ratio, and then draw the line. A straight line that misses $(0, 0)$ does not show a proportional relationship, so its point above $x = 1$ is not a unit rate.

Next, you will write and solve equations for proportional relationships in word problems, and you will analyze errors in work on proportional relationships.