Curve Sketching In Calculus
Step-by-Step Guide w/ 6 Examples!

Colonel Mustard, in the conservatory, with the candlestick!

Have you ever played the board game “Clue” by Hasbro?

The premise is that a person has been murdered in one of the rooms of a mansion, and it is up to the players to gather evidence and deduce the correct murder suspect, room, and weapon used for the dastardly deed.

While it’s a super fun game to play, you may be wondering how this has to do with curve sketching?

Just as in Clue, your job is to piece together important clues and facts to solve the crime. With curve sketching, you must identify essential characteristics of a function to produce a curve sketch.

So, what would be considered the “important” characteristics for sketching a curve?

Summary of Curve Sketching Techniques

Using the checklist above, we can sketch a curve while identifying the critical characteristics and components along the way.

Step-by-Step Example

For example, suppose we are asked to analyze and sketch the graph of the function.

\begin{equation}
f(x)=-\frac{1}{3} x^{3} + x -\frac{2}{3}
\end{equation}

Function Analysis

First, we will focus on our function analysis and find intercepts, domain, range, symmetry, asymptotes, and continuity.

\begin{equation}
\begin{array}{|r|l|}
\hline \text { Domain } & \begin{array}{l}
\text { All real numbers } \\
x \in(-\infty, \infty)
\end{array} \\
\hline \text { Range } & \begin{array}{l}
\text { All real number } \\
y \in(-\infty, \infty)
\end{array} \\
\hline \text { Symmetry } & \text { None } \\
\hline \text { Asymptotes } & \text { None } \\
\hline \text { Continuity } & \text { Continuous for all real numbers } \\
\hline
\end{array}
\end{equation}

First Derivative Analysis

Now we will concentrate on the first derivative to find critical numbers, increasing or decreasing intervals, as well as relative extrema.

\begin{equation}
\begin{array}{|r|l|}
\hline \text { First Derivative } & f^{\prime}(x)=-x^{2}+1 \\
\hline \text { Critical Numbers } & \begin{array}{l}
f^{\prime}(x)=0 \\
-x^{2}+1=0 \\
x=1,-1
\end{array} \\
\hline \text { Number Line } & \text { For } x<-1, f^{\prime}(x) \text { is negative } \\
& \begin{array}{l}
\text { For }-1<x<1, f^{\prime}(x) \text { is positive } \\
\text { For } x>1, f^{\prime}(x) \text { is negative }
\end{array} \\
\hline \text { Increasing Intervals } & x \in(-1,1) \text { because } f^{\prime}(x)>0 \\
\hline \text { Decreasing Intervals } & x \in(-\infty,-1) \cup(1, \infty) \text { because } f^{\prime}(x)<0 \\
\hline \text { Relative Maximum } & (1,0) \text { because } f^{\prime}(x) \text { changes from positive to negative } \\
\hline \text { Relative Minimum } & \left(-1,-\frac{4}{3}\right) \text { because } f^{\prime}(x) \text { changes from negative to positive } \\
\hline
\end{array}
\end{equation}

Second Derivative Analysis

Next, we will investigate the second derivative to points of inflection and concavity.

\begin{equation}
\begin{array}{|r|l|}
\hline \text { Second Derivative } & f^{\prime \prime}(x)=-2 x \\
\hline \begin{array}{r}
\text { Critical Numbers for the } \\
\text { Second Derivative }
\end{array} & \begin{array}{l}
f^{\prime \prime}(x)=0 \\
-2 x=0 \\
x=0
\end{array} \\
\hline \text { Number Line } & \begin{array}{l}
\text { For } x<0, f “(x) \text { is positive } \\
\text { For } x>0, f^{\prime \prime}(x) \text { is negative }
\end{array} \\
\hline \text { Points of Inflection } & \left(0,-\frac{2}{3}\right) \text { because } f^{\prime \prime}(x) \text { changes from positive to negative } \\
\hline \text { Concave Up Intervals } & x \in(-\infty, 0) \text { because } f^{\prime \prime}(x)>0 \\
\hline \text { Concave Down Intervals } & x \in(0, \infty) \text { because } f^{\prime \prime}(x)<0 \\
\hline
\end{array}
\end{equation}

Final Sketch

So, if we put all of this information together, we can generate the following graph.

In this video, we will become detectives, just like in the game of Clue, analyzing the graph of various functions and generating some pretty amazing sketches.

Alright, let’s get curve sketching!

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.