Topic 5.8 Notes – Sketching Graphs of Functions and Their Derivatives

How Derivatives Determine the Shape of a Graph

Think of it in layers:

• $f(x)$ → position (the graph itself)
• $f'(x)$ → slope of the graph
• $f''(x)$ → how the slope is changing

From that, almost everything follows.

What $f'$ tells you

• If $f'(x) > 0$, the function is increasing.
• If $f'(x) < 0$, the function is decreasing.
• If $f'(x) = 0$ or undefined (and $f$ exists), that’s a critical point.
• If $f'$ changes sign:
  • $+\to-$ → local maximum
  • $-\to+$ → local minimum

What $f''$ tells you

• If $f''(x) > 0$, the graph is concave up (cup shape).
• If $f''(x) < 0$, the graph is concave down (cap shape).
• If $f''$ changes sign → point of inflection.

Here’s how those ideas connect between a function and its derivative. In the diagrams, notice how the turning points of the original graph line up with where the derivative crosses the x-axis, and how positive and negative regions of $f'$ match increasing and decreasing behavior.

On AP questions, you’re often asked to justify conclusions using this logic, not just state them.

The Complete Graph Sketching Checklist

When you’re given a formula for $f$, build the graph in a logical order.

1. Domain and Continuity

• Polynomials → domain is all real numbers.
• Rational functions → denominator cannot be zero.
• Check for holes or vertical asymptotes.

If the function isn’t defined somewhere, that affects everything else.

2. Intercepts and Symmetry

• x-intercepts solve $f(x)=0$.
• y-intercept compute $f(0)$.
• Even symmetry: $f(-x)=f(x)$
• Odd symmetry: $f(-x)=-f(x)$

These give anchor points before calculus even starts.

3. Critical Points from $f'$

1. Find $f'(x)$.
2. Solve $f'(x)=0$.
3. Include where $f'$ is undefined but $f$ exists.

These x-values divide the graph into intervals for sign testing.

4. Increasing and Decreasing

Pick a test value in each interval between critical points.

• If $f'>0$ → increasing
• If $f'<0$ → decreasing

That sign change classifies extrema automatically.

5. Concavity and Inflection Points

1. Find $f''(x)$.
2. Solve $f''(x)=0$ or undefined.
3. Test signs on either side.

No sign change means no inflection point, even if $f''=0$. That’s a common trap.

First vs Second Derivative Tests

You have two ways to classify extrema.

First Derivative Test

Look at the sign change of $f'$.

• Works every time if signs are clear.
• Especially useful on FRQs where you must justify behavior.

Second Derivative Test

At a critical point where $f'=0$:

• $f''>0$ → local minimum
• $f''<0$ → local maximum
• $f''=0$ → inconclusive

Faster, but only works if $f''\neq 0$.

Reading Information from Graphs of $f'$ and $f''$

This shows up constantly on quizzes and the AP exam.

If you’re given the graph of $f'$

When you’re handed a graph of $f'$, your job is to translate what you see into behavior of $f$.

From a graph like this:

• Where $f'$ is above the x-axis → $f$ increasing.
• Where below → $f$ decreasing.
• Where $f'$ crosses the axis → possible extrema of $f$.
• If it just touches and turns around → horizontal tangent, not necessarily an extremum.

Students often confuse where $f'=0$ with where $f=0$. Totally different ideas.

If you’re given the graph of $f''$

• Above x-axis → concave up.
• Below x-axis → concave down.
• Sign change → inflection point.

Remember, inflection requires a sign change, not just touching zero.

Connecting All Representations

AP questions love mixing representations:

• Analytical: formulas for $f$, $f'$, $f''$
• Numerical: tables of values for $f'$
• Graphical: graphs of $f'$ or $f''$

You should be able to move between them smoothly.

Example from a table:
If a table shows $f'$ changing from negative to positive at $x=3$, you can justify that $f$ has a local minimum at $x=3$. That justification language matters on FRQs.