The Big Idea
The three graphs , , and are not three unrelated curves. They describe different features of the same function.
The original function tells you the height of the graph. Its first derivative tells you the slope of that graph. Its second derivative tells you how that slope is changing, which is the graph's concavity.
This gives you a powerful AP Calculus strategy: instead of trying to recognize a graph by its overall appearance, look for the relationships between the three graphs.
Think of the graphs as a chain: controls increasing/decreasing behavior, while controls how that increasing/decreasing behavior changes. In other words, is the slope of , and is the slope of .
The Most Important Connections
There are several relationships you should know almost automatically on the AP exam.
| Look at... | It tells you... | Key graphing relationship |
|---|---|---|
| Whether is increasing or decreasing | above the x-axis means is increasing. | |
| Local maxima and minima of | Zeros of can correspond to extrema of . | |
| Concavity of | above the x-axis means is concave up. | |
| Inflection points of | Zeros or sign changes of can identify inflection points. |
How to Read from
The first derivative is the slope of the original function. This is the single most important fact for matching with .
From to
- If is increasing, then its slope is positive, so .
- If is decreasing, then its slope is negative, so .
- If has a horizontal tangent, then its slope is zero, so .
What Happens at a Local Maximum?
Suppose rises toward a local maximum and then falls afterward.
Before the maximum, the function is increasing, so . After the maximum, the function is decreasing, so .
Therefore, must pass through zero at the local maximum.
Notice that does not merely have to equal zero. For a local maximum, the sign typically changes from positive to negative.
If changes from positive to negative at , then has a local maximum there. If changes from negative to positive, then has a local minimum.
What Tells You About Concavity
The second derivative is the derivative of the first derivative. So tells you whether the slopes of are getting larger or smaller.
That is exactly what concavity measures.
| Value of | What happens to slopes? | Shape of |
|---|---|---|
| Slopes are increasing | Concave up | |
| Slopes are decreasing | Concave down | |
| Possible change in concavity | Possible inflection point |
Inflection Points and
An inflection point occurs where the concavity of changes. Since measures concavity, we can use its sign to find these changes.
For example, if is positive to the left of and negative to the right, then the graph changes from concave up to concave down.
Importantly, simply finding is not enough. You need a change in concavity. That usually means a sign change in .
Do not automatically label every zero of as an inflection point. If touches zero but stays positive on both sides, the function remains concave up. There is no inflection point.
Worked Example 1: Matching and
Suppose the graph of has the following behavior:
- is decreasing on .
- reaches a local minimum at .
- is increasing on .
Worked Example 2: Matching and
Suppose is concave down for and concave up for .
Worked Example 3: Reading All Three Graphs Together
Suppose has a local maximum at , a local minimum at , and an inflection point at .
How to Identify the Three Graphs on an AP Question
When an AP Calculus question gives you three unlabeled graphs and asks you to determine which represents , , or , do not start by staring at their overall shapes.
Instead, use a systematic process.
- Find obvious extrema on the candidate graph. Those x-values should correspond to zeros of .
- Check increasing and decreasing intervals. Where increases, should be positive. Where it decreases, should be negative.
- Look at concavity. Where is concave up, should be positive. Where it is concave down, should be negative.
- Find inflection points. A change in concavity of should correspond to a sign change of .
- Use the derivative of the derivative. Since , the shape of must agree with the increasing/decreasing behavior of .
The Fast Matching Strategy
Zeros, Extrema, and Inflection Points
One of the most useful ways to organize these relationships is to recognize that a zero in one graph often appears as a special feature in another graph.
| Graph feature | What it can mean |
|---|---|
| Possible local maximum or minimum of | |
| changes sign | Local extremum of |
| Possible inflection point of | |
| changes sign | Inflection point of |
| has a local maximum | is zero there, when differentiable |
| has a local maximum | is zero there, when differentiable |
A Critical Distinction: Zero Is Not Always an Extremum
This distinction is especially important on AP Calculus multiple-choice questions.
If , that only tells you that the tangent to is horizontal. It does not automatically mean that has a local maximum or minimum.
For example, consider:
At , we have:
But is positive on both sides of zero, so the function is increasing on both sides. The graph has a horizontal tangent, but no local maximum or minimum there.
A critical point occurs where is zero or undefined. A local extremum requires additional information, such as an appropriate sign change in .
What the Shape of Tells You About
Because is the derivative of , you can reverse the usual reasoning. Instead of asking what tells you about , you can ask what the shape of tells you about .
- If is increasing, then is positive.
- If is decreasing, then is negative.
- If has a local maximum or minimum, then may be zero there.
- If is constant on an interval, then is zero on that interval.
What the Shape of Tells You About
The same logic continues one level higher.
Because is the derivative of :
- If is increasing, then .
- If is decreasing, then .
- If has a local maximum, then is typically zero there.
- If has a local minimum, then is typically zero there.
If a candidate graph has a local maximum at , then the corresponding graph should have a zero at . This is often much faster than trying to reconstruct the entire original function.
Practice Problems
If is increasing on an interval, what can you conclude about ?
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If changes from positive to negative at , what happens to ?
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If is negative on an interval, what does that tell you about the concavity of ?
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Suppose changes from negative to positive at . What can you conclude?
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A graph of has a local maximum at . What must be true about if is differentiable?
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You are given three candidate graphs. Graph A has zeros at and . Graph B has local extrema at exactly those same x-values. Which graph could represent if Graph B represents ?
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AP Exam Checklist
When you see a graphing question involving , , and , run through this checklist.
- Increasing/decreasing: Is increasing or decreasing? That determines the sign of .
- Extrema: Where does have local maxima or minima? Look for zeros and sign changes in .
- Concavity: Where is concave up or concave down? Look at the sign of .
- Inflection points: Where does change concavity? Look for a sign change in .
- Derivative shape: Remember that the increasing/decreasing behavior of is determined by the sign of .
This chain is the key to graph-matching problems. You do not need to know every coordinate of every graph. You need to know how their zeros, signs, extrema, and concavity must correspond.
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