Curve Sketching

Introduction to Curve Sketching

Again, Curve Sketching in Calculus makes us “appreciate the math” and helps visualize what is going on with function and their derivatives. It’s not my favorite part of calculus, but it’s a necessary evil :).

Before we get started, here are some basic graphs to look at. The graphs below show example original functions, and their derivatives (slopes). Notice how when we take the derivative, we go from a quadratic (2^nd degree) to a line (1^st degree), and from a cubic (3^rd degree) to a quadratic (2^nd degree). This makes sense, since the we are always going down a degree when we take a derivative. Also notice what happens where the derivative doesn’t exist (the last graph, an absolute value function).

Relative Extrema and the First Derivative Test

Extrema and Critical Numbers

We first talked about Extrema of Functions in the Advanced Functions section here. Extrema is just a fancy word for finding the lowest (minimum) or highest (maximum) $ \boldsymbol {y}$-value in a function or interval of a function; we will use Extrema to find Critical Numbers.

We can talk about absolute extrema, or relative (local) extrema. Think of the absolute extrema as the absolute lowest or highest point in the whole domain of the function, and the relative (local) extrema as the lowest or highest for a part of the graph. Technically, relative extrema must be the minimum or maximum of a point from both sides of $ \boldsymbol {x}$, so they can’t be endpoints; they are just “valleys” or “hills”. Note that not every function has a lowest (minimum) or highest (maximum) point in an interval or even the whole domain (like the function $ y=x$), so there may not be any extrema. The endpoints of a function may be the lowest or highest points (thus the absolute, not relative extrema); these are called the endpoint extrema.

When we find the minimum and maximum values in an interval, we can use this information to find where a function is decreasing and increasing, since at a minimum or maximum value, the function takes a turn from “down to up” or “up to down”.

Here is a graph that shows some examples of absolute/relative extrema; note also the endpoint extrema points:

Critical numbers or critical points exist where a function has a minimum or maximum, whether or not the function is differentiable at that point. And it turns out that if a function is differentiable at a certain point, and that point is a minimum or maximum, the derivative at that point is 0. Here is the formal definition of a critical number:

Increasing and Decreasing Functions, and the First Derivative Test

We talked about critical points (critical numbers) of a function (minimums or maximums), where the first derivative is 0 (or not defined). Now let’s talk about the derivative when the function is increasing (going upward from left to right), or decreasing (going downward from left to right).

When a function is increasing, the derivative (slope) is positive. When a function is decreasing, the derivative (slope) is negative. When a function is constant, or staying the same, the derivative (slope) is 0.

Here are the guidelines for finding intervals for which a function is increasing or decreasing: For a function $ f$ that is continuous on interval $ [a,b]$ and differentiable on interval $ (a,b)$, to find the intervals for which $ f$ is increasing or decreasing:

1. Find the critical points (minimums or maximums) in $ (a,b)$, and use these numbers to find test intervals.
2. For each of these test intervals, find the sign of the derivative at one test value.
3. If $ {f}’\left( x \right)>0$, then $ f$ is increasing on $ [a,b]$, if $ {f}’\left( x \right)<0$, then $ f$ is decreasing on $ [a,b]$, and if $ {f}’\left( x \right)=0$, then $ f$ is constant on $ [a,b]$.

Based on these guidelines, here is the First Derivative Test, which allows us to find relative minimums and maximums (also known as local minimums and local maximums).

Let’s think about why this makes sense. If we have a point where a function goes from falling to rising (negative to positive slope), that point would be a minimum. Similarly, if we have a point where a function goes from rising to falling (positive to negative slope), that point would be a maximum:

Here are some problems; notice that we are using sign charts to determine the intervals that the function is decreasing or increasing. Sometimes we need to use values that aren’t even in the domain of the function (like in vertical asymptotes) in the sign charts; theoretically, these aren’t critical numbers, since they don’t exist in the original function.


Here’s a First Derivative Problem with a trigonometric function:

Concavity and the Second Derivative

I like to think of concavity as “cup up” or “cup down”. Think of concave upwards of a cup that can hold water at all points, and concave downward is a cup that empties water out at all point. It turns out that when a graph is concave upward (cup up), its slope (first derivative) is increasing, so its second derivative is positive. When a graph is concave downward (cup down), its slope is decreasing, so its second derivative is negative.

A point of inflection (POI) is exactly where the concavity changes from concave up to concave down or concave down to concave up. It turns out that a graph crosses its tangent line at a POI.

Here’s an illustration of concavity:

The Second Derivative Test can also be used in curve sketching to find relative minima and relative maxima, and is the following:

You might see problems like this on concavity, points of inflection, and the Second Derivative Test. I think the best way to tackle these problems is to create a sign chart using points where the first derivative is 0 or undefined (critical values) and also where the second derivative is 0 or undefined:


Here’s a concavity problem with a trigonometric function:


Here are more Second Derivative Test problems:


Here’s one more problem where the Second Derivative Test does not apply at certain points:

Curve Sketching: General Rules

Here are some general curve sketching rules:

1. Find critical numbers (numbers that make the first derivative 0 or undefined).
2. Put the critical numbers in a sign chart to see where the first derivative is positive or negative (plug in the first derivative to get signs).
3. Where first derivative is positive, the function is increasing; where it’s negative, the function is decreasing (remember that you can combine two consecutive intervals only if the original function is defined for that critical number).
4. To get relative minimums and relative maximums, see how the derivative is changing. If it’s changing from negative to positive, it’s a minimum, and if it’s changing from positive to negative, it’s a maximum. To get the coordinates of the point at these places, plug the $ x$-value into the original function to get the $ y$-value.
5. Get the second derivative, and find the values where it’s either 0 or undefined. (Make sure these points are defined for both the function and first derivative).
6. Put those values in a sign chart to see where the second derivative is positive or negative (plug in the second derivative to get signs).
7. Where second derivative is positive, the graph is concave up, where the second derivative is negative, the graph is concave down. Remember that you can combine two consecutive intervals only if the original function is defined for that for the first and second derivative. A point of inflection (POI) occurs when the second derivate changes sign.
8. Use other points (can use a t-chart) to help graph!

Also, these tips may help:

1. If finding absolute extrema, find the critical numbers and endpoints, then plug into the original function to find $ y$-values. Compare all these values: the largest is the absolute maximum and the smallest is the absolute minimum.
2. When graphing, it might be helpful to identify any asymptotes or removable discontinuities (holes) by seeing what makes the denominator of the original function 0. Remember that a hole happens when you can cross out a factor in both the numerator and denominator (see Drawing Rational Graphs in the  Graphing Rational Functions, including Asymptotes section).

Here are some other hints that may help with the relationship of $ f$, $ {f}’$ and $ {{f}^{\prime \prime}}$. Again, remember that the relationship of $ f$ to $ {f}’$ is the same as $ {f}’$ to $ {{f}^{\prime \prime}}$; similarly, the relationship of $ {f}’$ to $ f$ is the same as $ {{f}^{\prime \prime}}$ to$ {f}’$. Also, I like to use the PMS acronym (sorry 🙂 ) to “travel” back and forth among the curves of the function, derivative, and second derivative:


Here’s a nice visual for curve sketching that shows the relationship of a function, derivative, and second derivative for the four curves shown:

Here are some problems that you may see:


Here’s one where we might have the information in a table.

Sketch a possible graph:


Here are more types of curve sketching problems you may see:


Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Optimization  – you are ready!