Curve Sketching Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Curve Sketching.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.

The first derivative tells you whether the function goes up or down (like reading a speedometer). The second derivative tells you whether it's speeding up or slowing down (like reading an accelerometer). Together, they give you a complete picture of the curve's shape.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The first derivative gives direction (up/down), the second gives concavity (bending up/down); together they shape the graph.

Common stuck point: The procedure for curve sketching is the easy part; the trap is assuming $f'(c)=0$ means an extremum. Asking "Am I using the signs of $f'$ and $f''$ to describe where the graph rises, turns, and bends?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I using the signs of $f'$ and $f''$ to describe where the graph rises, turns, and bends?

Worked Examples

Example 1

easy

Sketch

f(x) = x^3 - 3x^2

: find critical points, monotonicity, concavity, and inflection points.

Answer

Local max

(0,0)

; local min

(2,-4)

; inflection

(1,-2)

First step

f'(x) = 3x^2-6x = 3x(x-2)

. Critical points:

x=0, 2

Full solution

2
Increases on $(-\infty,0)$ , decreases on $(0,2)$ , increases on $(2,\infty)$ .
3
Local max $(0,0)$ , local min $(2,-4)$ .
4
$f''(x)=6x-6$ . Zero at $x=1$ ; sign changes: inflection at $(1,-2)$ .
5
Concave down on $(-\infty,1)$ , concave up on $(1,\infty)$ .

Systematic: find

f'

for critical points/monotonicity,

f''

for concavity/inflection. Use sign charts.

Example 2

hard

Sketch

f(x) = \dfrac{x^2}{x^2-1}

: find domain, asymptotes, critical points, and concavity.

Example 3

easy

Show the pattern: classify the critical point of

f(x)=x^2-2x+5

using the second derivative test.

Example 4

medium

For

f(x)=\frac{1}{x^2+1}

, find the maximum and describe the concavity at the maximum.

Example 5

medium

Why does the critical point

x=0

f(x)=x^4

produce a local minimum even though

f''(0)=0

Example 6

hard

Find the global maximum of

f(x)=x^2 e^{-x}

[0,4]

Example 7

challenge

Sketch all key features of

f(x)=\dfrac{x^2-4}{x^2-1}

: domain, asymptotes, intercepts, and critical points.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

For

f(x) = x^4 - 4x^3

, find and classify all critical points.

Example 2

medium

Find the inflection points of

f(x) = x^4 - 6x^2

Example 3

easy

On what interval is

f(x)=x^2

increasing?

Example 4

easy

Find the critical points of

f(x)=x^2-4x+1

Example 5

easy

f(x)=x^2

concave up or concave down?

Example 6

easy

Find

f''(x)

for

f(x)=x^3

Example 7

easy

Where is

f(x)=-x^2

decreasing?

Example 8

easy

Find the vertical asymptote of

f(x)=\frac{1}{x-3}

Example 9

easy

Classify the critical point of

f(x)=x^2-6x

using the second derivative test.

Example 10

easy

Find the

y

-intercept of

f(x)=x^3-2x+5

Example 11

medium

Find the inflection point of

f(x)=x^3-3x

Example 12

medium

Classify the critical points of

f(x)=x^3-3x

Example 13

medium

On what intervals is

f(x)=x^3-3x

increasing?

Example 14

medium

Find the concave-up interval of

f(x)=x^3-3x

Example 15

medium

Why does

f(x)=x^3

have a critical point at

0

that is not an extremum?

Example 16

medium

Find the horizontal asymptote of

f(x)=\frac{2x^2+1}{x^2-4}

Example 17

medium

A function is concave up and decreasing on an interval. Sketch its shape.

Example 18

medium

Find the absolute maximum of

f(x)=x^3-3x

[0,2]

Example 19

medium

Find the local extrema of

f(x)=x^4-2x^2

Example 20

challenge

Sketch the key features of

f(x)=\frac{x^2}{x^2+3}

: asymptote, extremum, concavity at

0

Example 21

challenge

Show

f(x)=x^4

has

f''(0)=0

but a local minimum at

0

Example 22

challenge

For

f(x)=xe^{-x}

, find the maximum and the inflection point.

Example 23

easy

Find the critical points of

f(x)=x^2-8x+3

Example 24

easy

Find

f''(x)

for

f(x)=x^4

Example 25

easy

Find the

y

-intercept and any

x

-intercepts of

f(x)=x^2-9

Example 26

easy

Find the horizontal asymptote of

f(x)=\frac{3x+1}{x-2}

Example 27

medium

Find the intervals where

f(x)=x^3-6x^2+9x

is increasing.

Example 28

medium

Find the inflection points of

concave down?

Example 39

challenge

For

f(x)=x^{1/3}(x-4)

, find the critical points and classify them.

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Related Concepts

Derivative Differentiation Rules Optimization

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativedifferentiation rulesoptimization