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Sketching Derivative Graphs from Function Curves: A Visual Guide
Learn to sketch a derivative graph by reading slope behavior—zero, positive, negative—from a given function curve, with practice tasks and a rubric.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
This guide teaches a visual skill: given the graph of a function, sketch the graph of its derivative without algebra. You will learn to read where the slope is zero, positive, or negative, and how steepness translates to height on the derivative graph. The method works for any smooth curve and builds intuition for calculus. All examples use simple grids and consistent units. A convention is defined for sketches: positive slope means the derivative graph is above the horizontal axis; negative slope means below; zero slope means the derivative graph touches or crosses the axis. Steepness is represented by vertical distance from the axis, using a consistent scale you choose and label. Because sketches are approximate, the rubric accepts any valid representation that preserves sign, zero locations, and relative steepness.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Reading Slope Behavior from a Function Curve
Start by marking key points on the function curve where the tangent is horizontal. These are the zeros of the derivative. Between these points, decide whether the curve is rising (positive slope) or falling (negative slope). On the derivative sketch, place points above the axis for rising intervals and below for falling intervals.
Next, estimate relative steepness. A steeper rise means a larger positive value; a gentler rise means a smaller positive value. Use a consistent vertical scale on the derivative grid, such as one grid square equals one unit of slope, and label it. The exact height is approximate, but the ordering of steepness should be preserved.
- Mark all visible points where the function curve has a horizontal tangent; these become x-intercepts on the derivative graph.
- For each interval between these points, note whether the function is increasing or decreasing.
- Sketch the derivative above the axis on increasing intervals and below on decreasing intervals.
- Adjust heights to reflect relative steepness, using a labeled consistent scale.
- Check that the derivative sketch changes sign only at the marked zeros and that its shape is smooth where the function is smooth.
Using a Consistent Sketch Convention
A sketch is not a precise plot, so define a convention before drawing. For example, let the horizontal axis represent x and the vertical axis represent slope value. Mark one grid square as one unit of slope (hypothetical example). This makes your sketch checkable: a point two squares above the axis means slope about +2.
Valid alternatives are allowed. You may use a different vertical scale as long as it is labeled and used consistently. You may draw the derivative as a smooth curve or as a series of connected line segments if the original curve has clear straight portions. The rubric checks sign, zero locations, and relative steepness, not exact pixel positions.
- Label the axes and state your vertical scale in the sketch margin.
- Use the same scale for all intervals in one sketch.
- Keep the derivative graph smooth where the function curve is smooth; use corners only where the function has a sharp corner.
- Indicate approximate heights with dots or short dashes if helpful, but keep the overall curve readable.
Common Patterns and Quick Checks
A straight line function with positive slope has a constant positive derivative, so its derivative graph is a horizontal line above the axis. A parabola opening upward has a derivative that is negative on the left, zero at the vertex, and positive on the right; the derivative graph is a line with positive slope. These patterns help you sanity-check your sketch.
Always verify that the number of derivative zeros matches the number of horizontal tangents you marked. If the function has a local maximum, the derivative changes from positive to negative; at a local minimum, it changes from negative to positive. Use these sign changes as checkpoints.
- Count horizontal tangents on the function curve and compare with x-intercepts on your derivative sketch.
- Check sign changes at each zero: maximum gives positive-to-negative, minimum gives negative-to-positive.
- Confirm that intervals of increasing function correspond to derivative above the axis.
- Confirm that intervals of decreasing function correspond to derivative below the axis.
Worked example
A function curve on a grid rises steeply from x=0 to x=2, reaches a flat top at x=2, then falls gently from x=2 to x=4. Sketch the derivative graph using a vertical scale where one grid square equals one unit of slope. Describe the sketch.
Worked answer
The derivative graph is above the axis from x=0 to x=2, with a high positive value near x=1 (about two squares up as a hypothetical example). At x=2 it crosses the axis (zero slope). From x=2 to x=4 it is below the axis, with a smaller negative value (about one square down) because the fall is gentler than the rise. The sketch is a smooth curve that starts positive, decreases to zero at x=2, and continues negative with a shallower depth.
Why: The function is increasing before x=2, so the derivative is positive; steeper rise means larger positive height. At x=2 the tangent is horizontal, so the derivative is zero. After x=2 the function decreases, so the derivative is negative; gentler fall means smaller magnitude. The sign change from positive to negative at x=2 matches a local maximum.
Check: Count horizontal tangents: one at x=2, matching one x-intercept. Check intervals: increasing gives above-axis, decreasing gives below-axis. Compare steepness: rise steeper than fall, so positive height greater in magnitude than negative height.
Four practice exercises
Exercise 1
A function curve is the straight line through the points (0,0) and (2,4) on a grid where one square equals one unit horizontally and one unit vertically. Sketch its derivative graph on a grid with the same horizontal scale, using a vertical scale where one square equals one unit of slope. State the height of your derivative line as a hypothetical example.
Show answer and check
The derivative graph is a horizontal line above the axis at a constant positive height. Since the line rises 4 units over 2 horizontal units, the slope is 4 ÷ 2 = 2, so the derivative line sits two squares above the axis (hypothetical example using the stated scale).
Why: A straight line has constant slope, so the derivative is constant. The slope is (4 − 0) ÷ (2 − 0) = 2, which is positive, so the derivative line is horizontal and above the axis at height 2 on the chosen scale.
Check: Verify the derivative is horizontal and above the axis. Recompute the slope as rise ÷ run = 4 ÷ 2 = 2 and confirm the line is drawn two squares up. Confirm the height is labeled and consistent with your chosen scale.
Exercise 2
A function curve is a parabola opening upward with its vertex at (1,0). It passes through (0,1) and (2,1), falling for x<1 and rising for x>1. Sketch the derivative graph on a grid where one square equals one unit horizontally and one unit vertically, and describe its shape and zero location.
Show answer and check
The derivative graph is a straight line with positive slope, crossing the axis at x=1. It is below the axis for x<1 and above for x>1. Using a labeled vertical scale where one square equals one unit of slope (hypothetical example), the derivative is about −2 at x=0 and about +2 at x=2, so the derivative sketch is a line through (1,0) rising 4 slope units over 2 horizontal units, from −2 at x=0 to +2 at x=2.
Why: The parabola has a horizontal tangent at its vertex x=1, so the derivative is zero there. The function decreases before x=1 (negative slope) and increases after (positive slope). For a parabola, the slope changes at a constant rate, so the derivative is linear.
Check: Check one x-intercept at x=1. Confirm the derivative is negative left of x=1 and positive right of it. Confirm the derivative line has positive slope and passes through (1,0). Verify that any heights you drew are labeled as approximate and consistent with your chosen scale.
Exercise 3
A function curve is made of three straight segments meeting at sharp corners: it rises from (0,0) to (1,1), then rises more steeply from (1,1) to (2,4), then is flat from (2,4) to (3,4). Sketch the derivative graph on a grid where one square equals one unit horizontally and one unit vertically, using a labeled vertical scale, and describe the heights on each interval.
Show answer and check
On (0,1) the slope is (1 − 0) ÷ (1 − 0) = 1, so the derivative is positive and one square up. On (1,2) the slope is (4 − 1) ÷ (2 − 1) = 3, so the derivative is positive and three squares up. On (2,3) the slope is 0, so the derivative is zero. Because the segments meet at sharp corners at x=1 and x=2, the derivative is undefined there; show a gap or vertical jump at each and label it undefined. The sketch otherwise shows a step up in positive height, then a drop to zero.
Why: Increasing intervals give positive derivative values. The steeper segment has the larger slope, so its derivative height is greater. The flat segment has zero slope, so the derivative is zero there. Sharp corners at x=1 and x=2 mean the derivative is undefined at those x-values.
Check: Confirm the derivative is above the axis on (0,2) and zero on (2,3). Check that the height on (1,2) is greater than on (0,1) by comparing slopes 3 and 1. Count one flat section matching one zero interval. Verify that x=1 and x=2 are marked as undefined (gap or vertical jump) because the segments meet at sharp corners there.
Exercise 4
A function curve has a local maximum at x=2 and a local minimum at x=5. It rises before x=2, falls between x=2 and x=5, and rises after x=5. On a grid where one square equals one unit horizontally, sketch the derivative graph using a labeled vertical scale, and mark its zeros and sign changes.
Show answer and check
The derivative crosses the axis at x=2 from positive to negative, and at x=5 from negative to positive. It is above the axis before x=2, below between x=2 and x=5, and above after x=5. Heights are approximate; for example, the positive height before x=2 may be about two squares up and the negative dip between x=2 and x=5 about one square down (hypothetical example).
Why: At a local maximum the slope changes from positive to negative, so the derivative changes from positive to negative. At a local minimum the slope changes from negative to positive. These sign changes determine the derivative graph's position relative to the axis.
Check: Verify two x-intercepts at x=2 and x=5. Check the sign pattern: positive, negative, positive. Confirm the sign changes match the maximum and minimum, and that your vertical scale is labeled and used consistently.
Common mistakes
- Confusing the height of the function curve with the height of the derivative graph. Focus on slope, not y-value. A high point on the function can have zero slope, so the derivative there is zero, not high.
- Drawing the derivative as a scaled copy of the function curve. The derivative graph shows slope values. Mark zeros where the function is flat, and place the derivative above or below the axis based on rising or falling.
- Ignoring relative steepness and drawing all positive slopes at the same height. Use a labeled vertical scale and vary the height to show steeper versus gentler slopes, preserving order.
- Placing derivative zeros at the x-coordinates of the function's peaks and valleys but forgetting flat plateaus. Mark every horizontal tangent, including flat sections, as a zero of the derivative.
Self-check rubric
- Derivative zeros are placed at x-values where the function curve has horizontal tangents.
- Derivative is above the axis on intervals where the function is increasing and below where it is decreasing.
- Relative steepness is preserved: steeper function slopes correspond to larger derivative magnitudes, using a labeled consistent scale.
- Sign changes at derivative zeros match local maxima (positive to negative) and local minima (negative to positive).
- Sketch is smooth where the function is smooth, and undefined points are marked where the function has sharp corners.
- Valid alternative scales or segment styles are accepted if labeled and internally consistent.
Questions and answers
Do I need to compute the derivative formula to sketch its graph?
No. This method uses only visual slope behavior: where the function is flat, rising, or falling, and how steep it is. The sketch is approximate but checkable.
How exact must the heights on the derivative sketch be?
Heights are approximate. Use a labeled consistent scale and preserve relative steepness. The rubric checks sign, zero locations, and ordering of steepness, not exact values.
What if the function curve has a sharp corner?
At a sharp corner the derivative is undefined. Show a gap or a vertical jump in the derivative sketch at that x-value, and note it as undefined.
Can I use straight line segments for the derivative sketch?
Yes, if the original curve has clear straight portions or if you are approximating. Keep the sign and zero locations correct, and label your scale.