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Sketching Graphs of
Functions & Derivatives

Learn to read the story a function tells through its slopes, turning points, and concavity: then sketch \( f'(x) \) and \( f''(x) \) with confidence.

AP Calculus BC

What Does \( f'(x) \) Tell Us?

The derivative \( f'(x) \) measures the instantaneous rate of change of \( f(x) \). Graphically, it equals the slope of the tangent line at every point on the curve.

\( f'(x) > 0 \)
\( f \) is increasing: the curve rises left to right.
\( f'(x) < 0 \)
\( f \) is decreasing: the curve falls left to right.
\( f'(x) = 0 \)
Horizontal tangent: possible max, min, or saddle.

Connecting \( f \) and \( f' \) Visually

f(x) max increasing decreasing f'(x) f' = 0 + −

What Does \( f''(x) \) Tell Us?

The second derivative \( f''(x) \) describes concavity: whether the curve bends upward or downward. It also measures the rate of change of the slope itself.

\( f''(x) > 0 \)
Concave up (cup shape). Slopes are increasing.
\( f''(x) < 0 \)
Concave down (cap shape). Slopes are decreasing.
\( f''(x) = 0 \)
Possible inflection point where concavity changes.

The Core Relationships

1
Zeros of \( f' \) correspond to horizontal tangents on \( f \): local maxima, local minima, or saddle points.
2
Sign changes in \( f' \) pinpoint where \( f \) switches between increasing and decreasing.
3
Extrema of \( f' \) (where \( f'' = 0 \)) correspond to inflection points of \( f \).
4
Where \( f' \) is increasing, \( f \) is concave up. Where \( f' \) is decreasing, \( f \) is concave down.

Reading \( f \) to Sketch \( f' \) and \( f'' \)

f(x) max min infl. f'(x) + − + f''(x) − +

Critical Point vs Inflection Point

Critical Point Inflection Point
Definition \( f'(c) = 0 \) or undefined \( f''(c) = 0 \) with sign change
On \( f \) Horizontal tangent Concavity reversal
On \( f' \) \( f' \) crosses the x-axis \( f' \) has a local extremum
On \( f'' \) No requirement \( f'' \) crosses the x-axis
Intuition Top of a hill or bottom of a valley Steepest point on an S-curve

How to Sketch \( f' \) from \( f \)

1
Mark zeros of \( f' \): Find every point where \( f \) has a horizontal tangent (peaks, valleys, flat spots). Plot these as x-intercepts on the \( f' \) axis.
2
Determine sign: Where \( f \) rises, \( f' > 0 \) (above x-axis). Where \( f \) falls, \( f' < 0 \) (below x-axis).
3
Locate extrema of \( f' \): Where the slope of \( f \) is steepest (inflection points), \( f' \) reaches a peak or valley.
4
Assess shape: If \( f \) is concave up, \( f' \) is increasing. If concave down, \( f' \) is decreasing. Connect the dots with a smooth curve.

Example 1: \( f(x) = x^2 - 4x + 3 \)

f(x) 1 2 3 4 min at \( x=2 \) f'(x) f' = 0 − +

Example 2: \( f(x) = x^3 - 3x \)

f(x) max min infl. f'(x) + − +

Example 3: \( f(x) = \sin(x) \)

\( f(x) = \sin(x) \) 0 \(\frac{\pi}{2}\) \(\pi\) \(\frac{3\pi}{2}\) \(2\pi\) \( f'(x) = \cos(x) \)

Going Backwards: Sketching \( f \) from \( f' \)

On the AP exam you may be given \( f' \) and asked to describe or sketch \( f \). Reverse the logic:

1
Where \( f' > 0 \), draw \( f \) going up. Where \( f' < 0 \), draw \( f \) going down.
2
Zeros of \( f' \) become turning points on \( f \). Check the sign change for max vs min.
3
Extrema of \( f' \) become inflection points on \( f \).
4
Where \( f' \) is increasing, \( f \) is concave up. Where \( f' \) is decreasing, \( f \) is concave down.

The Second Derivative Test

At a critical point \( c \) where \( f'(c) = 0 \):

\( f''(c) > 0 \)
\( f \) is concave up at \( c \), so \( c \) is a local minimum.
\( f''(c) < 0 \)
\( f \) is concave down at \( c \), so \( c \) is a local maximum.

Summary Cheat Sheet

Feature of \( f \) What \( f' \) Shows What \( f'' \) Shows
Increasing \( f' > 0 \) —
Decreasing \( f' < 0 \) —
Local max \( f' = 0 \), + to − \( f'' < 0 \)
Local min \( f' = 0 \), − to + \( f'' > 0 \)
Concave up \( f' \) is increasing \( f'' > 0 \)
Concave down \( f' \) is decreasing \( f'' < 0 \)
Inflection point \( f' \) has an extremum \( f'' = 0 \) with sign change

Common Mistakes to Avoid

Zero means extremum
\( f'(c) = 0 \) does not guarantee a max or min. Check for a sign change (e.g., \( f(x)=x^3 \) at \( x=0 \)).
Confusing f and f'
A high point on the \( f' \) graph means steepest slope, not a maximum of \( f \). Heights of \( f' \) are slopes of \( f \).
Ignoring concavity
Two curves can share zeros and sign pattern yet differ in shape. Always check whether \( f' \) is increasing or decreasing between zeros.

Question 1 of 5

If \( f \) has a local maximum at \( x = c \), what must be true about \( f'(c) \)?

Question 2 of 5

On the graph of \( f' \), you see a local minimum. What does this correspond to on the graph of \( f \)?

Question 3 of 5

If \( f'(x) > 0 \) and \( f''(x) < 0 \) on an interval, then \( f \) is:

Question 4 of 5

The graph of \( f' \) crosses the x-axis from negative to positive at \( x = 2 \). What can you conclude about \( f \) at \( x = 2 \)?

Question 5 of 5

Given \( f(x) = x^4 - 4x^3 \), at \( x = 0 \) we have \( f'(0) = 0 \) and \( f''(0) = 0 \). Using the First Derivative Test, what happens at \( x = 0 \)?