5.8 - Sketching Graphs of Functions and Their Derivatives

The Fundamental Connection

The slope of $f(x)$ at any point equals the y-value of $f'(x)$ at that same x-coordinate.

\text{Slope of } f \text{ at } x = c \quad \Leftrightarrow \quad y\text{-value of } f'(x) \text{ at } x = c

That relationship is what lets you sketch either graph from the other.

From $f(x)$ to $f'(x)$ : Reading Slopes

Find critical points where the slope is zero (horizontal tangents)

Label the slope signs along the curve: positive, negative, or zero

Estimate slope magnitude: steep sections give large values, flat sections give values near zero

Plot $f'(x)$ using those slopes as y-values

Example: Sketching $f'(x)$ from $f(x)$

5.8 - Sketching Graphs of Functions and Their Derivatives

Location	Slope Behavior	Value for $f'(x)$
Left end	Negative, increasing toward zero	Negative, approaching 0
Local min	Horizontal tangent	$f'(x) = 0$
Between min and max	Positive, rises then falls	Positive, with a maximum at the inflection point
Local max	Horizontal tangent	$f'(x) = 0$
Between max and right	Negative, most negative at the inflection point	Negative, with a minimum at the inflection point
Right end	Negative, increasing toward zero	Negative, approaching 0

Key observations:

$f'(x) = 0$ at the local extrema of $f(x)$

$f'(x)$ has extrema at the points of inflection of $f(x)$

Where $f$ is increasing, $f'(x) > 0$ (above the axis)

Where $f$ is decreasing, $f'(x) < 0$ (below the axis)

From $f'(x)$ to $f(x)$ : Building the Original Function

When you’re given the graph of

f'(x)

, look for:

Where $f'(x) = 0$ : these are critical points of $f$

Sign changes of $f'(x)$ :

Negative to positive means a local minimum of $f$

Positive to negative means a local maximum of $f$

Where $f'(x)$ has extrema: these are points of inflection of $f$

The slope of $f'(x)$ , which is $f''(x)$ :

$f'(x)$ increasing means $f''(x) > 0$ means $f$ is concave up

$f'(x)$ decreasing means $f''(x) < 0$ means $f$ is concave down

Example: Sketching $f(x)$ from $f'(x)$

Location on $f'(x)$	Feature of $f'(x)$	What it means for $f(x)$
First zero	Changes negative to positive	Local minimum
Maximum of $f'(x)$	$f''(x) = 0$ , changes + to -	Point of inflection
Second zero	Changes positive to negative	Local maximum
Minimum of $f'(x)$	$f''(x) = 0$ , changes - to +	Point of inflection
Third zero	Changes negative to positive	Local minimum

Important: The Vertical Shift Problem

When you sketch

f(x)

from

f'(x)

, there’s vertical ambiguity.

f(x) = 3x^2 - 2 \quad \Rightarrow \quad f'(x) = 6x

g(x) = 3x^2 + 1000 \quad \Rightarrow \quad g'(x) = 6x

Both have the same derivative but are shifted 1002 units apart vertically.

So your sketch can slide up or down anywhere. What actually has to be right:

The x-coordinates of the extrema

The x-coordinates of the inflection points

The concavity

The increasing/decreasing behavior

Any vertical shift is acceptable.

Pattern Recognition Shortcuts

$f(x)$	$f'(x)$	$f''(x)$
Cubic	Parabola	Linear
Parabola (like $x^2$ )	Linear (like $2x$ )	Constant (like $2$ )
Linear	Constant	Zero
Sine wave	Cosine wave	Negative sine wave

Example chain:

f(x) = x^2 \quad \rightarrow \quad f'(x) = 2x \quad \rightarrow \quad f''(x) = 2

Use with caution: these shortcuts work for standard forms but they won’t help with unusual functions. Always be ready to fall back on analyzing signs and behavior directly.

Complex Example: A Non-Standard Function

From this

f'(x)

, for

f(x)

Local minimum at $x_1$ (sign changes - to +)

Local maximum at $x_3$ (sign changes + to -)

Local minimum at $x_5$ (sign changes - to +)

The cusps of $f'(x)$ at $x_2$ and $x_4$ are where $f''(x)$ doesn’t exist, but they can still be inflection points if concavity changes

No memorized pattern helps here, you just have to analyze the behavior directly.

Using a Table of $f'$ and $f''$ Values

Sometimes you don’t get a graph at all, just values of

f'

and

f''

at a few points. You can still describe

f

at those points, as long as you only claim what the table actually shows.

Example:

f

is twice differentiable. The table gives selected values of its derivatives.

$x$	$-1$	0	2	4
$f'(x)$	5	0	$-4$	0
$f''(x)$	$-6$	$-4$	0	4

At $x = -1$ : $f'(-1) = 5 > 0$ , so $f$ is increasing there. Also $f''(-1) = -6 < 0$ , so $f$ is concave down there.

At $x = 0$ : $f'(0) = 0$ and $f''(0) = -4 < 0$ , so $f$ has a relative maximum at $x = 0$ by the second derivative test (5.7).

At $x = 2$ : $f'(2) = -4 < 0$ , so $f$ is decreasing there. $f''(2) = 0$ only makes $x = 2$ a candidate for a point of inflection. The table doesn’t show what $f''$ does right next to $x = 2$ , so you can’t confirm a sign change from the table alone.

At $x = 4$ : $f'(4) = 0$ and $f''(4) = 4 > 0$ , so $f$ has a relative minimum at $x = 4$ .

What a table can’t tell you: what happens between the listed points. Don’t claim whole intervals of increasing, decreasing, or concavity from a table unless the problem gives you more information, like a graph or a formula.

Using a Calculator to Graph Derivatives

TI-84 Method

Given

f(x) = \sin(e^x)

[0, 2.5]

Press Y=

Enter Y₁ = sin(e^x)

Move to Y₂

Press MATH, scroll to 8:nDeriv(, press ENTER

Syntax: nDeriv(Y₁, X, X)

First argument: the function (get it with VARS → Y-VARS → Function → Y₁)

Second argument: the variable you’re differentiating with respect to (X)

Third argument: the point to evaluate at (X means evaluate at every x-value)

Press GRAPH

The calculator plots both

f(x)

and

f'(x)

at once.

Verification:

Where $f(x)$ has a max or min, $f'(x) = 0$ (crosses the x-axis)

Where $f(x)$ is increasing, $f'(x) > 0$

Where $f(x)$ is decreasing, $f'(x) < 0$

Only use calculator methods when the problem allows them.

Key Takeaways

The slope of $f$ equals the y-value of $f'$ at the same x-coordinate

From $f$ to $f'$ : read slopes as y-values, and zeros happen at horizontal tangents

From $f'$ to $f$ : a zero of $f'$ with a sign change is a local extremum, and extrema of $f'$ are inflection points of $f$

Sign changes in $f'(x)$ : negative to positive is a minimum, positive to negative is a maximum

Vertical shifts don’t matter when sketching $f$ from $f'$ , any translation is valid

The x-coordinates of extrema and inflection points DO have to line up correctly

Pattern recognition (parabola → linear → constant) works only for standard forms

Concavity from $f'(x)$ : $f'$ increasing means $f$ concave up, $f'$ decreasing means $f$ concave down

Calculator shortcut: nDeriv(function, X, X) graphs the derivative

Back Next