Interpret and Check Graphs of Relationships

Start with a hike. Maya walks along a trail at a steady pace. Let $x$ be the time she has walked, in hours. Let $y$ be the distance she has walked, in miles. The distance depends on the time, so time is called the independent variable and distance is called the dependent variable.

In the figure below, the horizontal axis shows $x$ in hours and the vertical axis shows $y$ in miles. The grid lines are 1 unit apart, and the figure shows 0 to 10 on both axes. Check that spacing first whenever you read a graph, because it tells you what each grid line is worth.

Line A is the graph of the hike in that part of the plane. Point A and Point B are two points on Line A. Each one sits where two grid lines cross, so you can read its coordinates exactly.

Maya's hike can be shown in two more ways. An equation is a statement that two expressions are equal. Here it tells how to get the dependent variable, $y$, from the independent variable, $x$. Maya walks 2 miles each hour, so the distance is 2 times the time. The equation is $y = 2x$, where $2x$ means $2 \times x$.

A table lists some of the pairs. In the table below, each row is one ordered pair, so each row is one point on Line A. Every one of these points sits where two grid lines cross. The last column puts the row's $x$ into $2x$ and shows that the result equals the row's $y$.

Any of the three views can answer a question about the hike. Try this one: how long does Maya take to walk 8 miles?

• Graph: start at 8 on the vertical axis and go straight across to Line A. You meet it at Point B. Go straight down to 4 on the horizontal axis.
• Table: find the row where $y$ is 8. Its $x$ is 4.
• Equation: replace $y$ with 8 to get $8 = 2x$. Division undoes multiplication, so divide both sides by 2. Dividing both sides by the same nonzero number keeps them equal. Since $8 \div 2 = 4$ and $2x \div 2 = x$, you get $4 = x$.

All three give 4 hours, because all three describe the same relationship.

Now use that test to match an equation to a table and a graph. This time there is no story: $x$ and $y$ are two variables that stand for plain numbers with no units. The table below lists three pairs of their values. In the figure after it, the horizontal axis shows $x$, the vertical axis shows $y$, and the grid lines are 1 unit apart. The three pairs are plotted, which means marked as points on the plane. Each point is labeled with its coordinates, and Line B passes through all three.

When you graph an equation by hand, you work along a chain. The equation gives a table, the table gives points, and the points guide the line. A mistake breaks one link of the chain. To find it, check the links in order.

1. Equation to table. Put each $x$ into the equation. The result must equal the $y$ in the same row.
2. Table to points. Each row $(x, y)$ must be plotted at $x$ on the horizontal axis and $y$ on the vertical axis.
3. Points to line. When the graph of the equation is a straight line, as it is for every equation in this lesson, the points must all lie on one straight line, and the line you draw must pass through every one of them.

The first link that fails shows where a mistake was made.

The figure below shows a student's graph to check. Its horizontal axis shows $x$, its vertical axis shows $y$, and its grid lines are 1 unit apart.

Here is the idea to keep. An equation, a table and a graph are three views of one relationship. Each point $(x, y)$ on the graph of a real situation tells a small story: its first number is the value on the horizontal axis, its second number is the value on the vertical axis, and each comes with its unit. To connect the views or to check a graph, test pairs. A pair belongs only if it makes the equation true. When a graph has a mistake, check the chain from equation to table to points to line, and look for the first link that breaks.