What a Sign Chart Is For
A sign chart is one of the most useful organizational tools in AP Calculus. Instead of trying to reason about an entire function at once, you divide the number line into intervals and determine whether a quantity such as or is positive or negative on each interval.
The key idea is simple: the sign of a derivative tells you how the original function behaves. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. A sign chart makes those conclusions visible at a glance.
Think of as a "motion detector" for . A positive derivative means the graph is moving upward as increases; a negative derivative means it is moving downward.
The Basic Sign-Chart Method
To build a sign chart for a function's derivative, use this sequence:
- Find the derivative. Compute or, for concavity, .
- Find the critical numbers. Solve and identify values where does not exist, provided the original function is defined there.
- Put the critical numbers on a number line. These values divide the domain into intervals.
- Test the sign of the derivative on each interval. Usually one convenient test point is enough.
- Translate the signs into function behavior. Positive means increasing; negative means decreasing.
Why One Test Point Is Enough
Once you've identified every point where is zero or undefined, the sign of a continuous derivative cannot change inside one of the resulting intervals without passing through zero or becoming undefined.
That means you usually don't need to test every point. Choose one convenient value from each interval and evaluate the derivative there.
Worked Example 1: Increasing and Decreasing Intervals
Determine where is increasing and decreasing.
| Interval | Test point | Conclusion | |
|---|---|---|---|
| + | Increasing | ||
| − | Decreasing | ||
| + | Increasing |
The Sign Change Determines the Extrema
A critical number by itself does not automatically mean there is a local maximum or minimum. What matters is what the derivative does on either side.
| Before the critical number | At the point | After the critical number | Conclusion |
|---|---|---|---|
| + | 0 | − | Local maximum |
| − | 0 | + | Local minimum |
| + | 0 | + | No local extremum |
| − | 0 | − | No local extremum |
The first two rows are the important ones to memorize. A positive-to-negative change means the function goes from increasing to decreasing, producing a local maximum. A negative-to-positive change means it goes from decreasing to increasing, producing a local minimum.
Worked Example 2: A Critical Point That Is Not an Extremum
Analyze the function .
| Interval | Sign of | Behavior |
|---|---|---|
| + | Increasing | |
| + | Increasing |
Using Sign Charts for Concavity
The exact same strategy works with the second derivative.
Instead of asking whether is positive or negative, ask whether is positive or negative:
This gives you a second kind of sign chart:
| Sign of | Function behavior |
|---|---|
| + | Concave up |
| − | Concave down |
Worked Example 3: A Second-Derivative Sign Chart
Determine where is concave up and concave down.
| Interval | Test point | Conclusion | |
|---|---|---|---|
| + | Concave up | ||
| − | Concave down |
Factoring Makes Sign Charts Easier
When the derivative is factored, you can often determine its sign without calculating complicated decimal values. For example, suppose:
The critical numbers come directly from the factors. More importantly, the sign of each factor is easy to track as you move from left to right across the number line.
| Interval | Overall sign | ||
|---|---|---|---|
| − | − | + | |
| + | − | − | |
| + | + | + |
This factor-by-factor approach becomes especially useful when the derivative is a rational expression or a higher-degree polynomial.
Sign Charts and AP Free-Response Questions
On the AP exam, you may not always be asked to literally draw a sign chart. Instead, the reasoning behind one may appear in questions about increasing intervals, local extrema, concavity, or the number of solutions to an equation.
A strong response should connect the sign of the derivative to the requested conclusion. For example, don't simply write "maximum at ." Explain that changes from positive to negative, so the function changes from increasing to decreasing.
When justifying an extremum, explicitly describe the sign change. When justifying an inflection point, explicitly describe the change in concavity. The reasoning matters, not just the final coordinate.
Common Mistakes
A critical number is only a candidate. You must check whether the derivative changes sign. A change gives a local maximum; a change gives a local minimum.
The critical numbers tell you where the intervals split, but you need to test points inside the intervals to determine the derivative's sign there.
If a derivative is undefined at a value, that value may divide the domain even though it is not a zero of the derivative. Always consider the domain of the original function.
A positive means the function is increasing; it does not necessarily mean the function itself is positive. A function can be negative while increasing.
A Reliable Sign-Chart Checklist
- Find the derivative you need: for increasing/decreasing or for concavity.
- Find every zero and undefined value relevant to the derivative.
- Place those values in numerical order.
- Divide the number line into intervals.
- Choose one test point from each interval.
- Determine the sign of the derivative.
- Translate positive/negative signs into increasing/decreasing or concave-up/concave-down behavior.
- Look specifically for sign changes when identifying extrema or changes in concavity.
Practice Problems
Use a sign chart to determine where is increasing and decreasing.
Show answer
Use the first derivative to determine whether has a local maximum or minimum at .
Show answer
Use a sign chart for to determine where is concave up and concave down.
Show answer
For , use sign charts to determine the intervals of increase/decrease, local extrema, and concavity.
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon