Function Types and Concavity

Overview

This lecture covers how to determine if functions are linear, quadratic, or neither, using rates of change, and how to analyze concavity from tables of values.

Identifying Function Types from Tables

• For a linear function, the average rate of change is constant over any interval.
• For a quadratic function, the rate of change of the rate of change (second difference) is constant over consecutive equal length input intervals.
• When input intervals are not equal, always calculate the actual average rate of change to determine function type.
• If the average rate of change is not constant, the function is not linear.
• If the rate of change of the rate of change is not constant, the function is not quadratic.
• Use phrases like "over consecutive equal length input value intervals" to justify your answer for quadratic functions.

Practice Examples

• Example 1: Constant increase in rate of change indicates the function could be quadratic.
• Example 2: Variable input intervals and non-constant rate of change indicate the function is neither linear nor quadratic.
• Example 3: Constant average rates of change indicate the function could be linear, regardless of input interval lengths.
• Example 4: With equal input intervals but variable rate of change, function is neither linear nor quadratic.
• Example 5: Constant rate of change over equal intervals suggests the function could be linear.
• Example 6: Decreasing average rate of change at a constant rate over equal intervals suggests the function could be quadratic.

Determining Concavity

• A function is concave up where the rate of change is increasing over equal length intervals.
• A function is concave down where the rate of change is decreasing over equal length intervals.
• If average rate of change is constant, function is neither concave up nor concave down (most likely linear).
• If average rate of change increases and then decreases, function is neither concave up nor concave down.

Key Terms & Definitions

• Average Rate of Change — The change in output divided by the change in input over an interval.
• Linear Function — A function with a constant average rate of change.
• Quadratic Function — A function where the rate of change of the rate of change is constant over equal length intervals.
• Concave Up — Rate of change is increasing over equal length input intervals.
• Concave Down — Rate of change is decreasing over equal length input intervals.
• Consecutive Equal Length Intervals — Adjacent input intervals of the same size.

Action Items / Next Steps

• Practice identifying function types and concavity from tables as shown.
• Memorize the chart linking rate of change patterns to linear, quadratic, and concavity properties.