Sketching Curves Using Derivatives

Introduction

Welcome to the final lesson in our module on analyzing functions with derivatives. So far, we have built a powerful set of tools: the First Derivative Test for finding where a function is increasing or decreasing and locating local extrema, and the Second Derivative Test for determining concavity and identifying inflection points. This lesson is where we bring it all together. Our goal is to synthesize these individual techniques into a single, systematic process for one of the major applications of differential calculus: sketching an accurate graph of a function purely from its algebraic formula.

This process is like a diagnostic checklist. By methodically analyzing a function and its first two derivatives, we can uncover all the crucial features of its graph—peaks, valleys, points of changing curvature, and end behavior—and then assemble them into a complete visual representation. This corresponds to Section 4.5, "Summary of Curve Sketching," in your Stewart textbook.

A Systematic Approach to Curve Sketching

To avoid missing any key features of a graph, it's best to follow a consistent procedure. The following checklist covers all the essential information you need to gather before you start drawing.

Guidelines for Sketching a Curve $y = f (x)$

Domain: Determine the set of all possible input values $x$ for which the function is defined. This is especially important for rational functions (beware of division by zero) and functions with even roots (beware of negatives under the radical).
Intercepts:
- Find the y-intercept by evaluating $f (0)$ .
- Find the x-intercepts by solving the equation $f (x) = 0$ for $x$ , if possible.
Symmetry: Check for evenness ( $f (- x) = f (x)$ , symmetry about the y-axis), oddness ( $f (- x) = - f (x)$ , symmetry about the origin), or periodicity. This can reduce the amount of work needed.
Asymptotes:
- Horizontal Asymptotes: Evaluate $lim_{x \to \infty} f (x)$ and $lim_{x \to - \infty} f (x)$ .
- Vertical Asymptotes: Look for values of $x$ where the function is undefined, often where the denominator of a rational function is zero. Check the limits as $x$ approaches these values from the left and right.
Intervals of Increase/Decrease: Find the critical numbers by setting $f^{'} (x) = 0$ or finding where $f^{'} (x)$ is undefined. Use a sign chart for $f^{'} (x)$ to determine where the function is increasing ( $f^{'} (x) > 0$ ) and decreasing ( $f^{'} (x) < 0$ ).
Local Maxima and Minima: Use the First or Second Derivative Test to classify the critical numbers as local maxima or minima. Find the corresponding y-values for these points.
Concavity and Inflection Points: Find potential inflection points by setting $f^{''} (x) = 0$ or finding where $f^{''} (x)$ is undefined. Use a sign chart for $f^{''} (x)$ to determine where the graph is concave upward ( $f^{''} (x) > 0$ ) and concave downward ( $f^{''} (x) < 0$ ). Identify the coordinates of any inflection points.
Sketch the Curve: Plot the intercepts, critical points, and inflection points. Draw the asymptotes as dashed lines. Connect the points with a smooth curve that respects the intervals of increase/decrease and concavity.

To organize all this information, it's extremely helpful to use a table or a combined sign chart that shows the behavior of $f (x)$ based on the signs of $f^{'} (x)$ and $f^{''} (x)$ .

Example 1: Sketching a Polynomial

Let's begin with a polynomial function. These are often the most straightforward cases since their domain is all real numbers and they have no asymptotes. Professor Leonard's lecture provides an excellent, detailed walkthrough of the entire process.

Calculus 1 Lecture 3.6: How to Sketch Graphs of Functions

This video covers the complete curve-sketching process for a polynomial. Professor Leonard meticulously follows a step-by-step procedure, which aligns well with the checklist we just discussed.

Watch the segment from this example on sketching the function f(x) = (x+2)(x-1)^2. Follow along as he works through these key stages: Finding Intercepts: Notice how he finds both x- and y-intercepts first. Derivatives and Critical Points: He finds the first and second derivatives and their respective critical numbers and potential inflection points. The Master Table: Pay close attention to how he constructs the table (table setup) that combines the sign information from both derivatives. This is the crucial organizational step. Finding Coordinates: He finds the y-coordinates for all the interesting points (extrema, inflection points). Sketching: Finally, watch how he uses the table as a guide to draw the final graph, connecting the points and ensuring the shape (increasing/decreasing, concavity) is correct in each interval.

As you saw, the table provides a "blueprint" for the graph. It breaks the x-axis into intervals and tells you exactly what shape the curve has in each section:

Increasing and Concave Up
Increasing and Concave Down
Decreasing and Concave Up
Decreasing and Concave Down

Example 2: Sketching a Rational Function

Rational functions introduce the added complexity of asymptotes and domain restrictions. The sketching process remains the same, but we must pay careful attention to steps 1 (Domain) and 4 (Asymptotes).

Let's continue with Professor Leonard's video as he tackles a rational function. This example will reinforce the systematic approach while demonstrating how to handle these new features.

Calculus 1 Lecture 3.6: How to Sketch Graphs of Functions

In this part of the lecture, Professor Leonard sketches the rational function f(x) = \frac{x^2 - 1}{x^3}.

Please watch the following segments, focusing on how each step of our checklist is applied. Intercepts and Asymptotes: Watch from the start of the example (around this section) to see how he first establishes the domain, intercepts, and both vertical and horizontal asymptotes. This sets the "frame" for the graph. First and Second Derivatives: Next, observe his derivation of f'(x) and f''(x) from this segment. Note that for rational functions, finding critical numbers involves checking where the numerator and denominator of the derivative are zero or undefined. The Table and Final Sketch: Finally, see how he assembles the comprehensive table and uses it, along with all the points and asymptotes, to draw the final graph. The most instructive part of the sketching itself starts around this mark. Pay close attention to how the function's behavior near the asymptotes is determined by the information in the table.

Synthesizing the Information

The core of curve sketching lies in understanding the relationships between the signs of the first and second derivatives and the shape of the function's graph. The table below provides a concise summary of these connections. It's a useful reference to have as you practice these problems.

This table consolidates the meaning of the signs of $f'(x)$ and $f''(x)$. For example, a rising function has $f'(x) > 0$, and a concave up function has $f''(x) > 0$. A local minimum, where $f'(x)=0$, occurs in a concave up region where $f''(x) > 0$. Mastering these connections is the key to successful curve sketching.

Conclusion

In this lesson, we have unified all the analytical tools from this module into a comprehensive procedure for sketching functions. You learned to follow a systematic checklist to gather information about a function's domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, local extrema, and concavity. Most importantly, you saw how to organize this data into a table that serves as a blueprint for drawing an accurate and informative graph.

Key Takeaways:

Follow a Checklist: A systematic approach ensures no features are missed. The 8 steps (Domain, Intercepts, Symmetry, Asymptotes, Increase/Decrease, Local Extrema, Concavity, Sketch) provide a robust framework.
Organize with a Table: A combined sign chart for $f^{'} (x)$ and $f^{''} (x)$ is the most effective way to organize your findings before sketching.
Calculus Provides the "Why": While graphing software can instantly produce a graph, the methods learned here provide a deep understanding of why the graph has the shape it does.

This skill of analyzing and sketching functions is foundational. It prepares us for the next major topic: optimization. In our next lesson, we will begin Module 7 by learning how to find the absolute maximum and minimum values of a function over a specific interval, a direct application of the techniques you've just mastered.