Sketching Graphs: Functions & Derivatives

• New lesson editor
• MATH
• Secondary Education
• Age 12

Sketching Graphs: Functions & Derivatives

What do you already know about sketching graphs from functions and their derivatives?

Learning Objective

At the end of the lesson you will be able to sketch the graph of a function using its first and second derivatives, and understand how critical points, inflection points, and intervals of increase/decrease are related to the shape of the graph.

Graphs and Their Key Features

The graph of a function can show intervals where it increases or decreases, as well as points where it reaches a maximum, minimum, or changes curvature.

Derivatives and Graphs

Second Derivative (f'')

Indicates concavity: f'' > 0 means the graph is concave up, f'' < 0 means concave down. Inflection points occur where f'' = 0.

First Derivative (f')

Shows where the function is increasing (f' > 0) or decreasing (f' < 0). Critical points occur where f' = 0.

Finding Critical Points

Set the first derivative to zero (f'(x) = 0) to find critical points: possible maxima, minima, or stationary points.

Determining Inflection Points

Set the second derivative to zero (f''(x) = 0) to find potential inflection points, then test for a change in concavity.

Steps for Sketching Graphs

1. Find intercepts, critical points, and inflection points. 2. Determine intervals of increase, decrease, and concavity. 3. Use this info to sketch the function's graph.

Reflection & Practice

Practice

Sketch the graph for f(x) = x³ - 3x² + 2 using its first and second derivatives.

Recall

How does the first derivative help you sketch a function? Write your answer.

Write down 3 things you learned in this lesson.

Write down 2 things you want to know more about.

Ask 1 question about something you haven't quite understood yet.