Compare Cubic and Reciprocal Graph Shapes: Paired Plotting Worksheet

Algebra and Function Practice

This instructional set teaches side-by-side comparison of cubic and reciprocal graphs. You will plot each pair on one coordinate grid, then record how the cubic curve turns and crosses while the reciprocal curve splits into branches and approaches asymptotes. The convention for sketching is explicit: plot at least five labeled points per branch or curve, draw smooth curves through them, and mark asymptotes as dashed lines. Letter styles and handwriting vary, so the rubric rewards correct plotted coordinates, correct asymptote labels, and clear shape contrast rather than a single universal drawing style. All numerical values used in examples are hypothetical teaching values, not equipment specifications. Measure your own grid and ruler if you use physical materials.

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

Shared-grid plotting routine

For each pair, draw one coordinate grid and label both axes with equal unit spacing. Plot the cubic first using a small table of x-values that includes negative, zero, and positive inputs. Then plot the reciprocal using x-values that avoid zero, because division by zero is undefined. Use a dashed vertical line for the reciprocal asymptote at x = 0 and a dashed horizontal line for its horizontal asymptote at y = 0.

Keep the two curves visually distinct: label the cubic C and the reciprocal R. Do not connect across a vertical asymptote. A reciprocal branch on the left of the asymptote and a branch on the right are separate curves.

1. Choose x-values such as -3, -2, -1, 0, 1, 2, 3 for the cubic and -3, -2, -1, 1, 2, 3 for the reciprocal.
2. Compute y-values and record them in a table beside the grid.
3. Plot points, then draw smooth curves through them without crossing asymptotes.
4. Mark asymptotes with dashed lines and label turning points of the cubic.
5. Write one sentence comparing end behavior and one sentence comparing asymptote behavior.

Shape contrast: turning versus splitting

A cubic such as y = x^3 - 3x has turning points where its direction changes; it is a single continuous curve defined for all real x. A reciprocal such as y = 1/x has no turning point in the same sense; it has two separate branches and never touches either asymptote. On a shared grid, the cubic may cross the x-axis up to three times, while the reciprocal never crosses the x-axis because 1/x is never zero.

When comparing, state domain differences: the cubic domain is all real numbers, while the reciprocal domain excludes x = 0. State range differences where visible: the reciprocal range excludes y = 0. These statements are checkable from the plotted points and asymptote labels.

1. Identify whether the curve is one continuous piece or two branches.
2. Locate any turning point on the cubic and label its approximate coordinates.
3. Draw and label the reciprocal asymptotes.
4. Compare x-intercepts and y-intercepts using the plotted points.

Reading asymptote behavior from points

As x approaches zero from the right, 1/x becomes large positive; from the left, it becomes large negative. Plot points close to zero, such as x = 0.5 and x = -0.5, to show this separation. As x becomes large positive or large negative, 1/x approaches zero, so the horizontal asymptote is y = 0. The cubic does not approach a horizontal asymptote; its end behavior continues away from the axis.

Use simple arithmetic to check each plotted point. For y = 1/x, if x = 2 then y = 0.5; if x = -2 then y = -0.5. For y = x^3 - 3x, if x = 2 then y = 8 - 6 = 2; if x = -2 then y = -8 + 6 = -2. These checks confirm the plotted coordinates before drawing.

1. Compute reciprocal values near zero and far from zero.
2. Confirm that no reciprocal point lies on x = 0 or y = 0.
3. Check cubic values by substitution.
4. Label the asymptotes and write the domain and range exclusions.

Worked example

Compare y = x^3 - 3x and y = 1/x on one grid using x-values -3, -2, -1, 0, 1, 2, 3 where defined. Identify turning points and asymptotes.

Four practice exercises

Common mistakes

• Connecting the two reciprocal branches across the vertical asymptote. Draw each branch separately and leave a gap at x = 0; mark the asymptote as a dashed line.
• Plotting a reciprocal point at x = 0. Exclude x = 0 from the table because division by zero is undefined; use values such as 0.5 and -0.5 instead.
• Claiming the cubic has a horizontal asymptote because it looks flat near the turning point. Check end behavior with larger x-values; the cubic continues increasing or decreasing without approaching a fixed horizontal line.
• Using unequal axis spacing on the shared grid, which distorts shape comparison. Label equal unit intervals on both axes and measure the grid spacing before plotting.

Self-check rubric

• Plots at least five labeled points per curve or branch with correct coordinates.
• Draws smooth curves without connecting across a vertical asymptote.
• Labels vertical and horizontal asymptotes with dashed lines and equations.
• Identifies cubic turning points and states domain and range exclusions.
• Writes a clear comparison of shape, intercepts, and end behavior.
• Accepts valid alternative point choices and handwriting styles if coordinates and labels are correct.

Questions and answers