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Learn to graph lines directly from slope and intercept values. Includes worked example, common mistakes, and four practice tasks with answers and checks.
The equation y = mx + b gives two numbers that fully determine a straight line. The intercept b tells where the line crosses the y-axis, and the slope m tells how steeply it rises or falls. This lesson shows a reliable plotting routine: mark the intercept, use the slope as a rise over run, plot a second point, and draw the line. You will also see how to check your work by substituting coordinates back into the equation. All practice tasks are self-contained and use simple integer values so you can verify each step with arithmetic.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
In y = mx + b, the coefficient m is the slope and the constant b is the y-intercept. For example, in y = 2x + 3, m = 2 and b = 3. The line crosses the y-axis at (0, 3).
Slope is a ratio: rise over run. A slope of 2 means rise 2 units for every 1 unit run to the right. A slope of -1/2 means fall 1 unit for every 2 units run to the right. Keep the sign with the rise when moving right; alternatively, a negative slope can be plotted by moving left and up, but the standard routine moves right.
Always start at the y-intercept because it is given directly by b. Mark it clearly. Then use the slope to find a second point. If the slope is an integer, write it over 1, for example 3 = 3/1. If the slope is a fraction, use the numerator as the rise and the denominator as the run.
Example: for y = -2x + 1, b = 1 so plot (0, 1). The slope -2 = -2/1 means run 1 right and rise 2 down, giving (1, -1). A third check point is (2, -3). All three points should lie on the same straight line.
A point lies on the line exactly when its coordinates satisfy the equation. After plotting, pick one plotted point and substitute its x and y into y = mx + b. Both sides should be equal. This catches sign errors and swapped rise/run mistakes.
For y = 3x - 2, the point (1, 1) should satisfy 1 = 3(1) - 2, which is true. The point (2, 4) gives 4 = 3(2) - 2 = 4, also true. If a substituted point fails, recheck the slope direction and the intercept.
Negative slopes go downward as you move right. Keep the negative sign on the rise: for m = -3/4, run 4 right and rise 3 down. Fractional slopes with positive numerators go upward gradually; for m = 1/3, run 3 right and rise 1 up.
When b is negative, the intercept is below the x-axis, for example (0, -4). Plot it carefully and count from there. A common error is to treat a negative intercept as a positive one or to reverse the rise direction for negative slopes.
Plot the line y = -2x + 3 on a coordinate grid. Identify the intercept, the rise, and the run, and give two points on the line.
The y-intercept is b = 3, so plot (0, 3). The slope is m = -2 = -2/1, so run 1 right and rise 2 down from (0, 3) to reach (1, 1). A third point is (2, -1). Draw a straight line through (0, 3), (1, 1), and (2, -1). Check: for x = 1, y = -2(1) + 3 = 1, which matches (1, 1).
Plot the line y = 3x - 1. State the intercept, the rise, the run, and two points on the line.
Intercept (0, -1); slope 3 = 3/1, so rise 3 and run 1; second point (1, 2); third point (2, 5).
Why: b = -1 gives the intercept below the x-axis. The slope 3 means move 1 right and 3 up from (0, -1) to (1, 2), then again to (2, 5). Check: 3(1) - 1 = 2 and 3(2) - 1 = 5.
Check: Substitute x = 1 and x = 2 into y = 3x - 1 and confirm the y-values match the plotted points.
Plot the line y = -1/2 x + 2. State the intercept, the rise, the run, and two points on the line.
Intercept (0, 2); slope -1/2 means rise -1 and run 2; second point (2, 1); third point (4, 0).
Why: Start at (0, 2). Move 2 right and 1 down to (2, 1), then again to (4, 0). Check: -1/2(2) + 2 = 1 and -1/2(4) + 2 = 0.
Check: Substitute x = 2 and x = 4 into the equation and verify the results equal the y-coordinates.
Plot the line y = 4x + 0. State the intercept, the rise, the run, and two points on the line.
Intercept (0, 0); slope 4 = 4/1, so rise 4 and run 1; second point (1, 4); third point (2, 8).
Why: b = 0 means the line passes through the origin. From (0, 0), move 1 right and 4 up to (1, 4), then again to (2, 8). Check: 4(1) + 0 = 4 and 4(2) + 0 = 8.
Check: Substitute x = 1 and x = 2 into y = 4x and confirm the y-values are 4 and 8.
Plot the line y = -3x - 2. State the intercept, the rise, the run, and two points on the line.
Intercept (0, -2); slope -3 = -3/1, so rise -3 and run 1; second point (1, -5); third point (2, -8).
Why: Start at (0, -2). Move 1 right and 3 down to (1, -5), then again to (2, -8). Check: -3(1) - 2 = -5 and -3(2) - 2 = -8.
Check: Substitute x = 1 and x = 2 into y = -3x - 2 and confirm the y-values match the plotted points.
Write it as 5/1. Run 1 unit right and rise 5 units up. Plot a second point and draw the line.
Yes, but then reverse the rise sign. For m = 2/3, moving left 3 and down 2 gives another point on the same line. The standard routine moves right to keep signs simple.
Two points determine a line. Plotting a third point helps confirm the slope and catch errors before drawing.
The line passes through the origin (0, 0). Plot the origin, then use the slope to find a second point.