Sketching Graphs of
Functions & Derivatives
Learn to read the story a function tells through its slopes, connect \( f \) to \( f' \), and sketch one from the other with confidence.
AP Calculus AB
What Is a Derivative Graph?
The graph of \( f'(x) \) plots the instantaneous rate of change of \( f(x) \) at every point. Each \( y \)-value on the \( f' \) graph equals the slope of the tangent line to \( f \) at that same \( x \).
Seeing the Connection Visually
Why This Matters on the AP Exam
Free Response
You are given \( f' \) and asked to describe the behavior of \( f \): intervals of increase, decrease, and concavity.
Multiple Choice
Match a graph of \( f \) to the correct graph of \( f' \), or vice versa: a skill tested every year.
Justifications
Earn full credit by connecting sign of \( f' \) to increasing/decreasing and sign of \( f'' \) to concavity.
The Core Translation Rules
1
\( f \) increasing \( \Longleftrightarrow \) \( f'(x) > 0 \): the derivative graph is above the \( x \)-axis.
2
\( f \) decreasing \( \Longleftrightarrow \) \( f'(x) < 0 \): the derivative graph is below the \( x \)-axis.
3
Local max/min of \( f \) \( \Longleftrightarrow \) \( f'(x) = 0 \) and \( f' \) changes sign.
4
\( f \) concave up \( \Longleftrightarrow \) \( f' \) is increasing; \( f \) concave down \( \Longleftrightarrow \) \( f' \) is decreasing.
5
Inflection point of \( f \) \( \Longleftrightarrow \) \( f' \) has a local extremum (peak or valley).
Rules in Action: Diagram
Increasing vs. Concave Up: Key Distinction
| Increasing / Decreasing | Concave Up / Down | |
|---|---|---|
| What it describes | Direction of \( f \) | Curvature (bending) of \( f \) |
| Determined by | Sign of \( f'(x) \) | Sign of \( f''(x) \) or slope of \( f' \) |
| On the \( f' \) graph | \( f' \) above/below \( x \)-axis | \( f' \) increasing/decreasing |
| Special point | Local max/min where \( f' = 0 \) | Inflection where \( f' \) has extremum |
How to Sketch \( f' \) from \( f \)
1
Identify all local extrema of \( f \). Mark these \( x \)-values: they become the zeros of \( f' \).
2
Determine increasing/decreasing intervals. Where \( f \) rises, draw \( f' \) above the axis. Where \( f \) falls, draw \( f' \) below.
3
Judge steepness. Steeper slope on \( f \) means \( f' \) is farther from zero. Gentle slope means \( f' \) is near zero.
4
Check concavity. If \( f \) is concave up, \( f' \) should be increasing. If concave down, \( f' \) should be decreasing.
5
Connect smoothly. \( f' \) inherits continuity from \( f \) (if \( f \) is differentiable). Draw a smooth curve through your plotted features.
Example 1: \( f(x) = x^2 - 4x + 3 \)
Example 2: Sketching \( f' \) from a Cubic
Example 3: Sketching \( f \) from \( f' \)
Example 4: When \( f \) Has a Corner
Reading \( f \), \( f' \), and \( f'' \) Together
\( f \)
Position: where is the function \(f\)? Above or below the axis? What shape does the curve take?
\( f' \)
Velocity: is \( f \) going up or down? How fast? Zeros of \( f' \) are extrema of \( f \).
\( f'' \)
Acceleration: is \( f \) bending up or down? Zeros of \( f'' \) are inflection points of \( f \).
A continuous route through hills: elevation, grade, and intermediate height
For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.Height, grade and changing grade
For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.Common Mistakes to Avoid
Confusing \(f\) and \(f'\)
A peak on \( f \) does not mean a peak on \( f' \). It means \( f' = 0 \) there. The peak of \( f' \) corresponds to an inflection point of \( f \).
Ignoring Concavity
Getting increase/decrease right but drawing wrong curvature. Always check: is \( f' \) rising or falling in each interval?
Forgetting Non-differentiable Points
Corners, cusps, and vertical tangents mean \( f' \) is undefined there. Do not draw \( f' \) through these points.
Summary Cheat Sheet
| Feature of \( f \) | What \( f' \) does | What \( f'' \) does |
|---|---|---|
| Increasing | \( f' > 0 \) (above axis) | Any sign |
| Decreasing | \( f' < 0 \) (below axis) | Any sign |
| Local maximum | \( f' = 0 \), sign change \( + \to - \) | \( f'' < 0 \) |
| Local minimum | \( f' = 0 \), sign change \( - \to + \) | \( f'' > 0 \) |
| Concave up | \( f' \) is increasing | \( f'' > 0 \) |
| Concave down | \( f' \) is decreasing | \( f'' < 0 \) |
| Inflection point | \( f' \) has local extremum | \( f'' = 0 \) (sign change) |
Question 1 of 5
If \( f'(x) > 0 \) for all \( x \) in the interval \( (a, b) \), which of the following must be true about \( f \) on \( (a, b) \)?
When \( f'(x) > 0 \), the function is increasing. Concavity depends on \( f'' \), not \( f' \). The sign of \( f' \) tells us direction, not position above/below the axis. A local maximum requires \( f' \) to change sign, which cannot happen if \( f' > 0 \) throughout.
Question 2 of 5
The graph of \( f' \) crosses the \( x \)-axis from positive to negative at \( x = c \). What can you conclude about \( f \) at \( x = c \)?
When \( f' \) changes from positive to negative, \( f \) transitions from increasing to decreasing: that is a local maximum by the First Derivative Test.
Question 3 of 5
If \( f' \) has a local minimum at \( x = d \), what feature does \( f \) have at \( x = d \)?
An extremum of \( f' \) means \( f' \) changes from decreasing to increasing (or vice versa). That changes the concavity of \( f \), producing an inflection point: not a max or min of \( f \) itself.
Question 4 of 5
Suppose \( f(x) = x^4 - 4x^3 \). At which \( x \)-value does \( f \) have a local minimum?
We compute \( f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3) \). Setting \( f'(x) = 0 \) gives \( x = 0 \) and \( x = 3 \). At \( x = 0 \), \( f' \) does not change sign (it stays non-positive on both sides). At \( x = 3 \), \( f' \) changes from negative to positive, confirming a local minimum.
Question 5 of 5
The graph of \( f \) has a corner at \( x = k \). Which statement about \( f'(k) \) is correct?
At a corner, the left-hand derivative and the right-hand derivative are different finite values. Since they do not agree, the derivative \( f'(k) \) does not exist. The function is continuous but not differentiable there.