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Practice converting among tables, graphs, and equations for the same linear relationship, with checks for consistency and a clear rubric.
A linear relationship can be shown as a table of values, a straight-line graph, or an equation in the form y = mx + b. These three views are different pictures of the same rule. In this lesson you will move between them, check that they agree, and explain your reasoning. The convention for sketching graphs is simple: plot points at integer coordinates when possible, draw a straight line through them, and label the axes. Handwriting and letter styles vary, so the rubric focuses on correct values and clear labels rather than a single perfect style.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A table lists input-output pairs. For a linear relationship, the change in y divided by the change in x is constant. This constant is the slope m. Choose any two rows with different x-values, subtract the y-values and the x-values in the same order, and divide. Then use one point to find b by substituting into y = mx + b.
For example, a table shows (0, 3), (1, 5), (2, 7). The change in y is 2 when x increases by 1, so m = 2. Using (0, 3): 3 = 2(0) + b, so b = 3. The equation is y = 2x + 3. Check another row: when x = 2, y = 2(2) + 3 = 7, which matches the table.
From y = mx + b, the y-intercept b gives the point (0, b). The slope m tells how to move: from (0, b), go up m units for each 1 unit right if m is positive, or down if m is negative. Plot at least three points, including the intercept and one point using the slope, then draw a straight line through them.
Suppose y = -2x + 4. The intercept is (0, 4). With slope -2, move down 2 and right 1 to reach (1, 2), then (2, 0). Plot these three points and draw the line. A quick check: the point (2, 0) satisfies 0 = -2(2) + 4, which is true.
Choose x-values that are easy to read from the graph, such as 0, 1, and 2, or values where the line crosses grid intersections. Read the corresponding y-values from the graph. Record them in a table with columns for x and y. Then verify that the differences are constant; if they are not, re-read the points.
If a graph passes through (0, 1), (1, 3), and (2, 5), the table is x: 0, 1, 2 and y: 1, 3, 5. The change in y is 2 for each increase of 1 in x, so the slope is 2 and the intercept is 1. The equation is y = 2x + 1, which matches all three points.
After completing a table, graph, and equation, test one point from the table in the equation. Then confirm that the same point lies on your graph. If any view disagrees, find the error by recomputing the slope from two table rows, substituting a point into the equation, or re-reading a graph coordinate.
A useful check is to use the y-intercept. The table should contain (0, b), the graph should cross the y-axis at b, and the equation should show b as the constant term. If these do not match, the representations are not yet consistent.
A table shows (0, 2), (1, 6), (2, 10). Find the equation, describe the graph, and verify consistency.
Slope m = (6 - 2)/(1 - 0) = 4. Using (0, 2): 2 = 4(0) + b, so b = 2. Equation: y = 4x + 2. The graph is a straight line crossing the y-axis at (0, 2) and passing through (1, 6) and (2, 10). Check: when x = 2, y = 4(2) + 2 = 10, which matches the table.
A table shows (0, 5), (1, 8), (2, 11). Write the equation and describe the graph.
y = 3x + 5. The graph is a straight line crossing the y-axis at (0, 5) and passing through (1, 8) and (2, 11).
Why: Change in y is 3 for each increase of 1 in x, so m = 3. At x = 0, y = 5, so b = 5.
Check: Substitute x = 2 into y = 3x + 5 to get 11, which matches the table.
An equation is y = -x + 4. Build a table for x = 0, 1, 2 and describe the graph.
Table: (0, 4), (1, 3), (2, 2). The graph is a straight line with slope -1 crossing the y-axis at (0, 4).
Why: Substitute each x into the equation: 0 gives 4, 1 gives 3, 2 gives 2. The slope is -1, so the line goes down 1 for each 1 right.
Check: Plot the three points; they should lie on a straight line. Verify that (1, 3) satisfies 3 = -1 + 4.
A graph (described) passes through (0, 1), (1, 4), and (2, 7). Write the table and equation.
Table: x = 0, 1, 2; y = 1, 4, 7. Equation: y = 3x + 1.
Why: The y-intercept is 1. The change in y is 3 for each increase of 1 in x, so m = 3.
Check: Test (2, 7) in y = 3x + 1: 3(2) + 1 = 7, which matches.
A table shows (1, 2), (3, 8), (5, 14). Find the equation and check with a fourth point x = 4.
y = 3x - 1. When x = 4, y = 11.
Why: Slope m = (8 - 2)/(3 - 1) = 6/2 = 3. Using (1, 2): 2 = 3(1) + b, so b = -1.
Check: Substitute x = 4 into y = 3x - 1 to get 11. Also check (5, 14): 3(5) - 1 = 14.
You can still find b by substituting any point into y = mx + b after finding m. For example, if (2, 7) is on the line and m = 3, then 7 = 3(2) + b, so b = 1.
Yes. A negative slope means the line goes down from left to right. In a table, y decreases as x increases. In the equation, m is negative.
Two points determine a line, but plotting a third point helps catch errors. Use at least three points when possible.
Recheck the table calculations. A linear relationship must have a constant rate of change. If the differences are not constant, the relationship is not linear.