Graphing Linear Functions Worksheet: Slope-Intercept Practice

Algebra and Function Practice

Graphing a line from slope-intercept form is a repeatable procedure: identify b as the y-intercept, use m as a rise-over-run step, plot a second point, then draw the straight line through both points. This lesson explains that procedure with clear conventions, a worked example, common mistakes, and four self-contained practice tasks. All coordinates use the same unit scale, and every answer can be checked by substitution.

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

Reading y = mx + b

In y = mx + b, the value b is the y-coordinate where the line crosses the y-axis, so the point (0, b) is always on the line. The value m is the slope: it tells how much y changes for each one-unit increase in x. A slope of 2 means rise 2 for run 1; a slope of -1/2 means fall 1 for run 2.

Keep units consistent. If each grid square represents one unit on both axes, then a rise of 3 means three squares up and a run of 2 means two squares right. Label the scale on each grid before plotting so the visual step matches the numerical step.

1. Write the equation in y = mx + b form and circle b.
2. Plot the y-intercept at (0, b).
3. Write m as a fraction rise/run, keeping the sign on the rise.
4. From the intercept, move run units right and rise units up or down; plot the second point.
5. Draw a straight line through both points and extend it across the grid.
6. Verify by substituting the second point's coordinates into the equation.

Choosing a reliable second point

A second point is reliable when it lies far enough from the intercept that small plotting errors are visible. Moving one run step is acceptable, but moving two or three run steps gives a better check. For m = 3/4, from (0, 2) move 4 right and 3 up to (4, 5); substituting gives 5 = 3/4(4) + 2, which is correct.

If the slope is negative, the rise is negative. For m = -2/3, from (0, 1) move 3 right and 2 down to (3, -1). Substitution gives -1 = -2/3(3) + 1, which checks. Always place the negative sign with the rise, not the run, to avoid reversing direction.

1. Convert integer slopes to fractions: m = 4 becomes 4/1.
2. Keep the negative sign on the numerator when the slope is negative.
3. Move right for positive run; the run is always positive in this convention.
4. Plot the second point and label its coordinates.
5. Substitute both coordinates into the original equation to confirm equality.

Drawing and checking the line

Use a straightedge so the line passes exactly through both plotted points. Extend the line to the edges of the grid and add arrows only if the grid represents all real numbers. Label the line with its equation near one end so the graph is self-documenting.

A quick visual check: the line should cross the y-axis exactly at b and should rise from left to right when m is positive, fall when m is negative, and stay horizontal when m = 0. A vertical line is not a function and is outside this task.

1. Align a straightedge through both points before drawing.
2. Extend the line across the full grid.
3. Confirm the y-intercept visually matches b.
4. Confirm the direction matches the sign of m.
5. Substitute the second point to complete the numerical check.

Worked example

Graph y = (2/3)x - 1 on a grid where each square is one unit. Identify the intercept, plot a second point, and verify it.

Four practice exercises

Common mistakes

• Plotting b on the x-axis instead of the y-axis. The intercept is (0, b); start at the origin and move only vertically to reach it.
• Applying run before rise or reversing rise and run. Write m as rise/run and move horizontally by the denominator first, then vertically by the numerator.
• Dropping the negative sign when the slope is negative. Keep the sign on the rise: for m = -3/2, move 2 right and 3 down.
• Drawing a curved or bent line through the points. Use a straightedge; a linear function must be perfectly straight.

Self-check rubric

• Y-intercept plotted correctly at (0, b).
• Slope interpreted as rise over run with correct sign.
• Second point plotted accurately and labeled.
• Straight line drawn through both points with a straightedge.
• Verification by substitution shown or stated.
• Line extended across the grid and labeled with its equation.

Questions and answers