Topic 5.8 Notes – Sketching Graphs of Functions and Their Derivatives

How derivatives determine the shape of a graph

Everything starts with what the derivatives mean.

First derivative $f'(x)$ → slope behavior

• $f'(x) > 0$ → $f$ is increasing
• $f'(x) < 0$ → $f$ is decreasing
• $f'(x) = 0$ or undefined → critical point

Critical points are candidates for local maxima or minima. They are not automatically extrema.

Second derivative $f''(x)$ → concavity

• $f''(x) > 0$ → concave up (slopes increasing)
• $f''(x) < 0$ → concave down (slopes decreasing)
• Sign change in $f''$ → possible inflection point

Here’s a single graph that ties these ideas together. Notice how the sign of $f'$ matches increasing or decreasing behavior, and how points where $f' = 0$ or is undefined show up as critical points.

Graph showing critical points and intervals of increase and decrease

If you can look at a graph like this and mentally describe where $f'$ is positive or negative and where $f''$ changes sign, you understand this topic.

The full graph sketching process

When you’re given an equation and asked to sketch $f$, build the structure piece by piece.

1. Domain and discontinuities

• Polynomials → domain is all real numbers.
• Rational/log functions → check where undefined.
• Note vertical asymptotes or holes if they exist.

2. Intercepts and symmetry

• x‑intercepts → solve $f(x)=0$
• y‑intercept → compute $f(0)$
• Even → $f(-x)=f(x)$
• Odd → $f(-x)=-f(x)$

These anchor your sketch.

3. Critical points

Solve $f'(x)=0$ (or where undefined but in domain).

These split the number line into intervals for sign analysis.

4. Increasing and decreasing intervals

Test values in each interval:

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

If the sign changes:

• $+ \to -$ → local max
• $- \to +$ → local min

You must reference the sign change when justifying on FRQs.

5. Concavity and inflection points

Solve $f''(x)=0$ for candidates.
Check sign change of $f''$.

No sign change means no inflection point. This is a very common mistake.

Moving between $f$, $f'$, and $f''$

A lot of AP questions give you only a graph of $f'$ or a table of values of $f''$ and ask about $f$. You need to translate quickly.

Here’s the relationship summary:

One key connection students miss:

If $f'$ has a local max or min, then $f''=0$ there.
Extrema of $f'$ correspond to zeros of $f''$.

Using numerical and graphical information

Sometimes you’ll get a table:

From just this:

• $f'$ changes from negative to positive at $x=2$
• So $f$ has a local minimum at 2

Even without an equation.

If you’re given a graph of $f'$ that touches but doesn’t cross the axis, that means:

• $f'(c)=0$
• No sign change
• No local extremum

The AP loves that subtlety.

Justifying correctly

Strong explanations sound like this:

• “Since $f'(x) > 0$ for $1<x<4$, $f$ is increasing on $(1,4)$.”
• “Because $f'$ changes from positive to negative at $x=3$, $f$ has a local maximum at 3.”
• “Since $f''(x) < 0$, the graph is concave down.”

Always reference the derivative, the sign, and the interval or point.