Complete a Table and Graph a Proportional Relationship

To complete a table, first find $k$ from a row where you know both numbers and $x$ is not $0$: $k = y \div x$. Every other row with $x$ not $0$ must give that same result when you divide $y$ by $x$.

• To find a missing $y$-value, multiply: $y = k \cdot x$. Since $y \div x$ must equal $k$, the $y$-value has to be $k$ times the $x$-value.
• To find a missing $x$-value, divide: $x = y \div k$. This undoes the multiplication, and you may do it whenever $k$ is not $0$.
• In the row where $x = 0$, the $y$-value is $0$, because $k \cdot 0 = 0$. You cannot test this row by dividing, because you cannot divide by $0$, but $y = kx$ still gives its $y$-value.
• In the row where $x = 1$, the $y$-value is $k$, because $k \cdot 1 = k$. This row shows the unit rate.

To graph the relationship, write each row as a point $(x, y)$ and plot it. When $x$ can be any amount from $0$ upward, such as a time or a weight, draw the straight line through the points, starting at $(0, 0)$.

See how the graph matches the table. Each row became one point, and each label is that row written as $(x, y)$, with minutes first and liters second.

• $(0, 0)$ is the origin: after $0$ minutes the tank holds $0$ liters. The graph of every proportional relationship contains this point, because $k \cdot 0 = 0$.
• $(1, 3)$ shows the unit rate: $3$ liters in $1$ minute. Its $y$-value is the constant $k = 3$.
• All five points are on Line A because every pair fits $y = 3x$. Each extra minute adds the same $3$ liters, so the points rise at one steady steepness and line up.

You know the points line up from the arithmetic, not from how the picture looks.

The graph above shows the same mistake. As before, minutes are on the horizontal $x$-axis and liters are on the vertical $y$-axis. The points $(0, 0)$, $(1, 3)$ and $(2, 6)$ are on Line A, and so is the correct point $(4, 12)$, because $3 \cdot 4 = 12$. The hollow point $(4, 9)$ is the mistaken row. It has the same $x$-value as $(4, 12)$ but a smaller $y$-value, so it sits below the line.

Only one straight line passes through $(0, 0)$ and $(2, 6)$. That line is Line A, and at $x = 4$ it has $y = 12$, not $9$. So $(0, 0)$, $(2, 6)$ and $(4, 9)$ do not lie on one straight line, and a table with those rows is not proportional. When a relationship is supposed to be proportional, a point that falls off the line through $(0, 0)$ and the other points tells you to recheck that row.

Takeaway: in a proportional relationship, one constant $k$ builds everything. Because $y \div x = k$ for every pair with $x$ not $0$, you have $y = kx$. Multiply by $k$ to fill in a missing $y$-value, divide by $k$ (when $k$ is not $0$) to fill in a missing $x$-value, and write $0$ beside $x = 0$. Plot each row as a point, and the points lie on one straight line through $(0, 0)$, with $(1, k)$ showing the unit rate. Next, you will write an equation for a proportional relationship straight from its table or its graph.