Graphs of Functions and Derivatives Worksheet: Verify Slope-Sign Correspondence

Algebra and Function Practice

This instructional set teaches students to compare a function graph with a proposed derivative graph by checking slope-sign correspondence. Students do not sketch derivatives from scratch; instead, they verify whether the derivative graph correctly shows where the function rises, falls, and has horizontal tangents. The convention for reading slope sign is explicit: a function is increasing where its graph goes up from left to right, decreasing where it goes down, and has a horizontal tangent at a smooth local extremum. The derivative graph must be positive on increasing intervals, negative on decreasing intervals, and zero at extrema. Letter styles and handwriting vary, so the rubric accepts any clear notation that matches the stated convention. All resources are planned and free; they are not yet built.

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

Reading slope sign from a function graph

A function graph rises from left to right on an increasing interval. On that interval, the derivative must be positive. A function graph falls from left to right on a decreasing interval, so the derivative must be negative there. At a smooth local maximum or minimum, the function has a horizontal tangent, so the derivative must be zero at that x-value.

Use a simple sign table. Choose test x-values in each interval between critical points. Record whether the function graph goes up or down, then record whether the derivative graph is above or below the x-axis. The two records must agree.

1. Mark the x-values where the function graph has horizontal tangents or sharp corners.
2. List the intervals between these x-values.
3. In each interval, note whether the function graph rises or falls.
4. In the same interval, note whether the derivative graph is positive or negative.
5. Check that rising matches positive and falling matches negative.
6. At each marked x-value, check that the derivative graph crosses or touches zero as expected.

Checking extrema and zero crossings

A local maximum on the function graph should correspond to a zero on the derivative graph where the derivative changes from positive to negative. A local minimum should correspond to a zero where the derivative changes from negative to positive. If the derivative graph is zero but does not change sign, the function has a horizontal tangent but not necessarily an extremum.

Do not require the derivative graph to have the same y-values as the function. Only the sign pattern and zero locations matter for this comparison.

1. Locate each local maximum on the function graph.
2. Find the matching x-value on the derivative graph and confirm the derivative is zero there.
3. Confirm the derivative is positive just before that x-value and negative just after.
4. Repeat for each local minimum with the opposite sign change.
5. Reject a pair if any extremum lacks the required sign change in the derivative graph.

Using the worksheet checkboxes

The planned worksheet has four side-by-side function/derivative graph pairs. Each pair includes checkboxes for increasing intervals, decreasing intervals, local maxima, and local minima. Students tick a box only when the derivative graph agrees with the function graph at that feature. The answer key explains each tick or rejection.

A pair is valid only if every checkbox that should be ticked is ticked and no incorrect box is ticked. Partial credit may be given for correct interval checks even if an extremum check is wrong, but the pair as a whole is not fully verified.

1. For each pair, read the function graph from left to right.
2. Tick increasing intervals where the derivative is positive.
3. Tick decreasing intervals where the derivative is negative.
4. Tick local maxima where the derivative changes from positive to negative through zero.
5. Tick local minima where the derivative changes from negative to positive through zero.
6. Compare your ticks with the answer key explanations.

Handling valid alternative notations

Students may label intervals with inequality notation, interval notation, or a sign chart. All are acceptable if the meaning is clear and consistent with the stated convention. For example, writing x < 2 or (-infinity, 2) for an increasing interval is equally valid. The rubric checks the mathematical content, not the style.

When a graph has a sharp corner, the derivative is not defined there. In this task, treat sharp corners as points to note separately and do not require the derivative graph to be zero at a corner.

1. Choose one notation style and use it consistently.
2. Write intervals so that the left and right endpoints are clear.
3. Use open intervals for increasing/decreasing unless the convention explicitly includes endpoints.
4. Mark undefined derivative points separately from zero derivative points.

Worked example

A function graph rises on (-2, 1), falls on (1, 4), and has a local maximum at x = 1. A proposed derivative graph is positive on (-2, 1), negative on (1, 4), and zero at x = 1. Does the derivative graph correctly match the function graph?

Four practice exercises

Common mistakes

• Assuming the derivative graph must have the same shape as the function graph. Compare only slope sign and zero locations. The derivative graph shows steepness, not height.
• Ticking an increasing interval when the derivative graph is negative there. Re-read the function graph left to right. If it falls, the derivative must be negative.
• Treating every zero of the derivative graph as an extremum. Check for a sign change. A zero without a sign change is a horizontal tangent, not an extremum.
• Ignoring sharp corners on the function graph. Mark corners as points where the derivative is undefined and do not force a zero there.

Self-check rubric

• Increasing intervals on the function graph are matched to positive values on the derivative graph.
• Decreasing intervals on the function graph are matched to negative values on the derivative graph.
• Each local maximum is matched to a zero on the derivative graph with a positive-to-negative sign change.
• Each local minimum is matched to a zero on the derivative graph with a negative-to-positive sign change.
• Sharp corners are noted as points where the derivative is undefined, not forced to zero.
• Notation is clear and consistent; alternative valid notations are accepted.

Questions and answers