A Library of Parent Functions (Lesson 3.1)
Unit 3 Day 1
CED Topic(s): 1.12
Understand that the parent function represents the most basic function in a family of functions.
Describe the key features of six parent functions: identity, absolute value, square root, quadratic, cubic, and reciprocal.
Analyze and compare the key features of parent functions.
Quick Lesson Plan
This lesson helps students to review six of the parent functions that they have studied in previous years. We expect that students already know the basic features of these graphs, so we encourage them to complete the activity without the use of a calculator. Students may choose to fill out more or less ordered pairs on their table depending on their familiarity with the shape of the function. As you are monitoring students, you can ask students to compare the different parent functions, making note of similarities and differences. Students that are comfortable with the shapes of the graph and/or finish early could be challenged to come up with scenarios that could be modeled by each of these function families.
How is the absolute value function like a piecewise function?
What can you say about the rate of change of this [identity] function?
Why isn’t y=mx+b included in this list?
What can you say about the rate of change of y=x^2?
What can you say about the rate of change of y=x^3?
What can you say about the rate of change of y=sqrt(x)?
Which of these six functions are concave up? Which are concave down?
While this lesson is not formally included in the AP Precalculus course framework, we thought it was valuable to include before students explore transformations (Lesson 3.2) and determine an appropriate function model (Lesson 3.4). Understanding families of functions and how they behave is critical in this course and the six parent functions students look at today are a good starting toolkit. Exponential and trigonometric parent functions will be introduced throughout the course.
Although students have seen all of these functions before, some students may struggle to remember some of these key features, specifically for the reciprocal function. Note that the reciprocal function and the square root function are the only parent functions in this set with restricted domains, and the reciprocal function is the only one with a vertical asymptote. It is important that students understand the key features of the parent function before investigating the effect of transformations in subsequent lessons. Students also review intervals of increasing and decreasing from Lesson 1.2. The vocabulary of “identity function” may be new to students. Ask them to consider why it is called this. Make sure students understand that the identity function is the parent function of all linear functions.
The Check Your Understanding questions increase in rigor. The last question provides a good challenge for students as they consider how the outputs of the six parent functions differ over the interval from x=0 to x=1. The answer may go against students’ intuition about which functions grow faster. This is a good opportunity to review some elementary concepts of fractions and decimals, and later concepts of exponent properties.