Comparing Quadratic, Square Root, and Cube Root Functions Represented in Different Ways

Examples, solutions, and videos to help Algebra I students learn how to compare two different quadratic, square root, or cube root functions represented as graphs, tables, or equations. They interpret, contextualize and abstract various scenarios to complete the comparative analysis.

New York State Common Core Math Algebra I, Module 4, Lesson 22
Worksheets for Algebra 1

Opening Exercise

Populate the table on the right with values from the graph.

Briefly discuss ways to recognize key features in both representations of this function.
What is the vertex for the function? Find it and circle it in both the table and the graph.
What is the y-intercept for the function? Find it and circle it in both the table and the graph.
What are the x-intercepts for the function? Find them and circle them in both the table and the graph.

Exploratory Challenge/Exercises

2. At an amusement park, there is a ride called The Centre. The ride is a cylindrical room that spins as the riders stand along the wall. As the ride reaches maximum speed, riders are pinned against the wall and are unable to move.

The model that represents the speed necessary to hold the riders against the wall is given by the function s(r) = 5.05√ r, where s = required speed of the ride (in meters per second) and r = the radius (in meters) of the ride.

In a competing ride called The Spinner, a car spins around a center post. The measurements in the table below show the relationship between the radius (r) of the spin, in meters, and the speed (s) of the car, in m/sec.

Due to limited space at the carnival, the maximum spin radius of rides is 4 meters. Assume that the spin radius of both rides is exactly 4 meters. If riders prefer a faster spinning experience, which ride should they choose? Show how you arrived at your answer.

Lesson 22 Summary

The key features of a quadratic function, which are the zeros (roots), the vertex, and the leading coefficient, can be used to interpret the function in a context (e.g., the vertex represents the maximum or minimum value of the function). Graphing calculators and bivariate data tables are useful tools when comparing functions.

Lesson 22 Problem Set Sample Solutions

1. One type of rectangle has lengths that are always two inches more than their widths. The function f describes the relationship between the width of this rectangle in x inches and its area, f(x), in square inches and is represented by the table below.

A second type of rectangle has lengths that are always one-half of their widths. The function g(x) = 1/2 x² describes the relationship between the width given in x inches and the area, g(x), given in square inches of such a rectangle.

a. Use g(x) to determine the area of a rectangle of the second type if the width is 20 inches.

b. Why is (0, 0) contained in the graphs of both functions? Explain the meaning of (0, 0) in terms of the situations that the functions describe.

c. Determine which function has a greater average rate of change on the interval 0 ≤ x ≤ 3.

d. Interpret your answer to part (c) in terms of the situation being described.

e. Which type of rectangle has a greater area when the width is 5 inches? By how much?

f. Will the first type of rectangle always have a greater area than the second type of rectangle when widths are the same? Explain how you know.

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