Graphical Curve Analysis and Behavior

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

View all topics

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory

Derivative behavior model

Curve analysis combines the first derivative, second derivative, and graph features to describe the complete behavior of a function.

Step 1. Use the first derivative for increasing and decreasing behavior.

\[ \begin{aligned} f'(x) &> 0 &&\Rightarrow \text{the function is increasing} \\ f'(x) &< 0 &&\Rightarrow \text{the function is decreasing} \end{aligned} \]

Step 2. Find critical points.

Critical points occur inside the interval where the derivative is zero or undefined.

\[ \begin{aligned} f'(c) &= 0 \end{aligned} \]

A critical point is only a candidate. It may be a local maximum, local minimum, or neither.

Step 3. Use the second derivative for concavity.

\[ \begin{aligned} f''(x) &> 0 &&\Rightarrow \text{concave up} \\ f''(x) &< 0 &&\Rightarrow \text{concave down} \end{aligned} \]

Step 4. Find inflection points.

An inflection point is a point where the graph changes concavity.

\[ \begin{aligned} f''(c) &= 0 \end{aligned} \]

Important: \(f''(c)=0\) only gives an inflection candidate. Concavity must change across the point.

Complete curve-analysis checklist

A full graphical analysis usually includes the following steps.

Step 1. Identify the domain and possible discontinuities.

\[ \begin{aligned} \text{Domain} &= \{x:\ f(x)\ \text{is defined}\} \end{aligned} \]

Step 2. Find critical points from the first derivative.

\[ \begin{aligned} f'(x) &= 0 \end{aligned} \]

Step 3. Build increasing and decreasing intervals.

Test the sign of \(f'(x)\) on each interval.

\[ \begin{aligned} f'(x)>0 &\Rightarrow \text{increasing} \\ f'(x)<0 &\Rightarrow \text{decreasing} \end{aligned} \]

Step 4. Find inflection candidates from the second derivative.

\[ \begin{aligned} f''(x) &= 0 \end{aligned} \]

Step 5. Build concavity intervals.

Test the sign of \(f''(x)\) on each interval.

\[ \begin{aligned} f''(x)>0 &\Rightarrow \text{concave up} \\ f''(x)<0 &\Rightarrow \text{concave down} \end{aligned} \]

Step 6. Check asymptotes and end behavior.

Vertical asymptotes often occur where the function becomes undefined and grows without bound.

\[ \begin{aligned} x=a &\Rightarrow \text{possible vertical asymptote if } f(x)\to \pm\infty \end{aligned} \]

Worked example

Consider \(f(x)=x^3-3x\). This function is useful because it has both increasing/decreasing intervals and a change in concavity.

Step 1. Differentiate the function.

\[ \begin{aligned} f(x) &= x^3-3x \\ f'(x) &= 3x^2-3 \end{aligned} \]

Step 2. Find critical points.

\[ \begin{aligned} 3x^2-3 &= 0 \\ 3(x^2-1) &= 0 \\ x^2-1 &= 0 \\ x &= -1,\ 1 \end{aligned} \]

The function is increasing where \(f'(x)>0\) and decreasing where \(f'(x)<0\).

\[ \begin{aligned} f'(x)>0 &\quad \text{on }(-\infty,-1)\cup(1,\infty) \\ f'(x)<0 &\quad \text{on }(-1,1) \end{aligned} \]

Step 3. Use the second derivative for concavity.

\[ \begin{aligned} f''(x) &= 6x \end{aligned} \]

\[ \begin{aligned} f''(x)<0 &\quad \text{for }x<0 \\ f''(x)>0 &\quad \text{for }x>0 \end{aligned} \]

Therefore, the graph is concave down on \((-\infty,0)\), concave up on \((0,\infty)\), and has an inflection point at \(x=0\).

Common mistakes

Confusing \(f\), \(f'\), and \(f''\): \(f\) gives height, \(f'\) gives slope, and \(f''\) gives concavity.
Assuming \(f'(x)=0\) always means a maximum or minimum: it may be neither.
Assuming \(f''(x)=0\) always means inflection: concavity must actually change.
Ignoring discontinuities: rational, logarithmic, and square-root functions may have restricted domains.
Trusting asymptote detection blindly: numerical asymptote detection is approximate and should be checked algebraically when exact results are needed.

Frequently Asked Questions

How do you know where a function is increasing?

A function is increasing where f'(x)>0.

How do you know where a function is decreasing?

A function is decreasing where f'(x)<0.

How do you identify concavity?

The graph is concave up where f''(x)>0 and concave down where f''(x)<0.

What is a critical point?

A critical point occurs where f'(x)=0 or where f'(x) is undefined while f(x) exists.

What is an inflection point?

An inflection point is a point where the graph changes concavity, usually detected by a sign change in f''(x).

Can the calculator detect asymptotes?

It estimates vertical asymptotes numerically by looking for undefined values, very large values, or sudden jumps in the graph.

Are the derivative values exact?

The calculator uses numerical derivatives, so results are approximate and should be interpreted as graphical and computational estimates.

Can I show only some graph layers?

Yes. The graph has toggles for f(x), f'(x), f''(x), critical points, inflection points, asymptotes, labels, and grid.

Graphical Curve Analysis and Behavior

Function and analysis window

Multi-layer graph and output settings

Quick examples

Dynamic curve behavior graph

Step-by-step curve behavior analysis

Rate this calculator

Frequently Asked Questions

Function and analysis window

Multi-layer graph and output settings

Quick examples

Dynamic curve behavior graph

Step-by-step curve behavior analysis

Rate this calculator

Frequently Asked Questions

Related calculators