Graph Proportional Relationships from Ratio Tables: Plot, Draw, and Verify

Algebra and Function Practice

A ratio table lists pairs of quantities that scale together. When those pairs are proportional, plotting them on a coordinate grid produces points that lie on a straight line through the origin. This lesson shows how to read a ratio table, choose axes, plot the pairs, draw the line, and verify the constant of proportionality so the graph is consistent with the table.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Read the table and identify the constant

Each row or column of a ratio table gives an ordered pair. For proportional quantities, dividing the second quantity by the first gives the same value every time; that value is the constant of proportionality, often written k. If the table shows 2 and 6, then 6 divided by 2 equals 3. If it shows 5 and 15, then 15 divided by 5 equals 3. The constant is 3.

Before plotting, confirm the table is proportional by checking at least two pairs. If the quotients differ, the relationship is not proportional and should not be drawn as a straight line through the origin.

1. Copy each pair from the table as (first quantity, second quantity).
2. Divide the second quantity by the first for two or more pairs.
3. Confirm the quotients match; record that value as k.
4. Label the horizontal axis with the first quantity and the vertical axis with the second quantity.

Plot the pairs and draw the line

Plot each ordered pair by moving along the horizontal axis first, then the vertical axis. Mark each point clearly. Because the relationship is proportional, the points should line up. Place a straightedge through the plotted points and extend the line through the origin (0, 0). The origin is part of every proportional graph because zero of one quantity corresponds to zero of the other.

If a plotted point does not fall on the line formed by the others, recheck the table entry and the plotting steps before changing the line. A single misplotted point is more common than a non-proportional table.

1. Plot each ordered pair and label it with its coordinates.
2. Place a straightedge through the points and check they are collinear.
3. Extend the line to pass through the origin.
4. Choose one clear point on the line and confirm its quotient equals k.

Verify slope and constant of proportionality

The slope of a proportional line equals the constant of proportionality. To verify, pick two points on the drawn line, subtract the vertical values and the horizontal values in the same order, and divide. For example, using (2, 6) and (5, 15): vertical change is 15 minus 6 equals 9; horizontal change is 5 minus 2 equals 3; 9 divided by 3 equals 3, matching k.

A quick check is to read one point from the line, such as (4, 12), and divide 12 by 4 to get 3. If the result matches the table's constant, the graph and table agree.

1. Select two points on the line with clear coordinates.
2. Subtract vertical values and horizontal values in the same order.
3. Divide the vertical change by the horizontal change.
4. Compare the result with k from the table.

Worked example

A ratio table shows (1, 4), (3, 12), and (6, 24). Plot the pairs, draw the proportional line, and verify the constant of proportionality.

Four practice exercises

Common mistakes

• Plotting the pair in reverse order, so the first quantity is placed on the vertical axis. Label axes before plotting and always move horizontally first, then vertically, using the table's order.
• Drawing a line that does not pass through the origin. Extend the straightedge until the line crosses (0, 0); proportional graphs always include the origin.
• Assuming a table is proportional after checking only one pair. Check at least two pairs by dividing the second quantity by the first and confirming the quotients match.
• Using different subtraction orders when computing slope. Subtract vertical values and horizontal values in the same point order, then divide.

Self-check rubric

• Axes are labeled with the quantities from the table and a consistent scale.
• All ordered pairs from the table are plotted in the correct order.
• The drawn line is straight and passes through the origin.
• The constant of proportionality is computed from at least two pairs and stated clearly.
• The slope check uses two points on the line and matches the constant.
• Valid alternative axis scales or point labels are accepted if the plotted positions and slope check remain correct.

Questions and answers