Identify Proportional Graphs: Straight Lines Through the Origin

Algebra and Function Practice

A proportional relationship between two quantities has a constant ratio, and its graph is always a straight line passing through the origin (0,0). This lesson teaches you to inspect any graph and decide whether it represents a proportional relationship. You will check two features: is the graph a straight line, and does that line pass through the origin? If both are true, the relationship is proportional. If either fails, it is not. You will also learn to justify your judgment with evidence from the graph, such as specific points or the y-intercept. The convention for reading coordinates is standard: each point is (x, y), with x measured along the horizontal axis and y along the vertical axis. Numerical values used in examples are hypothetical and chosen for clarity; always measure actual graphs you are given.

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

The Two-Part Test for Proportional Graphs

A graph represents a proportional relationship exactly when it is a straight line that passes through the origin. This gives a simple two-part test. First, check whether the graph is a straight line. Curves, bent lines, or disconnected points fail this part. Second, check whether the line passes through (0,0). A straight line that crosses the y-axis at any value other than zero fails this part.

Both conditions must hold. A straight line through the origin means the ratio y/x is constant for every point on the line (except x=0, where the ratio is undefined but the line still passes through the origin). A straight line not through the origin has a nonzero y-intercept, so the ratio y/x changes from point to point. A curve can pass through the origin but still fail because the ratio is not constant.

When you inspect a graph, read at least two clear points to confirm the line is straight and to check the origin. If the line visibly passes through (0,0) and continues straight, the relationship is proportional. If the line is straight but crosses the y-axis at, say, (0,3), it is not proportional. If the graph is a curve through (0,0), it is not proportional.

1. Look at the overall shape: is it a single straight line?
2. Locate the origin (0,0) on the graph.
3. Check whether the line passes through the origin.
4. If both conditions hold, judge the relationship proportional; otherwise, judge it not proportional.
5. Record a justification using a specific feature, such as 'straight line through (0,0)' or 'straight line with y-intercept 3'.

Reading Points and Justifying Your Judgment

A strong justification names the feature that decides the case. For a proportional graph, say: 'The graph is a straight line and it passes through the origin.' For a non-proportional straight line, say: 'The graph is a straight line but it crosses the y-axis at (0, b) with b not zero.' For a curve, say: 'The graph is not a straight line.' You can also use two points to show the ratio is not constant.

For example, suppose a straight line passes through (0,2) and (4,6). The ratio at (4,6) is 6/4 = 1.5, but the line does not pass through the origin, so it is not proportional. Suppose a curve passes through (0,0), (1,1), and (2,4). The ratios are 1/1 = 1 and 4/2 = 2, which differ, so the relationship is not proportional even though the curve passes through the origin.

Use consistent units when reading axes. If the horizontal axis is time in seconds and the vertical axis is distance in meters, a point (3, 6) means 3 seconds and 6 meters. The ratio 6/3 = 2 meters per second is constant only if the graph is a straight line through the origin. Label any numerical values you choose as hypothetical examples, and measure the actual graph in front of you.

1. Identify two clear points on the graph and write their coordinates.
2. Compute y/x for each point where x is not zero.
3. Compare the ratios: equal ratios with a straight line through the origin support proportionality.
4. State the deciding feature in one sentence.

Common Traps: Straight but Offset, or Through Origin but Curved

Two traps catch many students. The first is a straight line that does not pass through the origin. It looks orderly, but the nonzero y-intercept means the ratio y/x is not constant. The second is a curve that passes through the origin. Passing through (0,0) is necessary but not sufficient; the graph must also be straight.

A third trap is a graph with only one point shown. One point cannot establish a proportional relationship because you cannot confirm a straight line through the origin from a single point. Look for a line or enough points to determine the shape.

Always separate the two checks. Ask: straight? Then ask: through origin? Keep the answers independent until you combine them. This prevents the common error of accepting any line through the origin or any straight line.

1. Check straightness first, independent of the origin.
2. Check origin passage second, independent of straightness.
3. Combine the two results: proportional only if both are yes.
4. If only one point is visible, state that more information is needed.

Worked example

A graph shows a straight line passing through (0,0) and (5,10). Judge whether the relationship is proportional and justify your answer.

Four practice exercises

Common mistakes

• Judging a straight line as proportional just because it is straight. Also check that the line passes through (0,0). A straight line with a nonzero y-intercept is not proportional.
• Judging a curve as proportional because it passes through the origin. The graph must also be a straight line. A curve through the origin is not proportional.
• Using only one point to decide. One point cannot show a straight line through the origin. Look for a line or at least two points that determine the shape.
• Computing y/x at a point where x is zero. The ratio y/x is undefined at x=0. Use points with nonzero x to compare ratios, and check origin passage visually.

Self-check rubric

• States a clear yes or no judgment.
• Checks both conditions: straight line and passage through the origin.
• Provides a justification naming a specific feature, such as 'straight line through (0,0)' or 'y-intercept not zero'.
• Uses correct coordinates and consistent units when citing points.
• Avoids claiming proportionality from a single point or from origin passage alone.

Questions and answers