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4.2

Activity

In Pattern A, the length and width of the rectangle grow by one small square from each step to the next.
In Pattern B, the number of small squares doubles from each step to the next.
In each pattern, the number of small squares is a function of the step number, .

<p>Four steps of a growing pattern. Step 0 is blank. Step 1: One square. Step 2: Four squares in a two by two square. Step 3: Nine squares in a three by three square.</p> — Pattern A

Write an equation to represent the number of small squares at Step in Pattern A.
Is the function linear, quadratic, or exponential?

Complete the table:

, step number	, number of small squares
0
1
2
3
4
5
6
7
8

<p>Four steps of a growing pattern. Step 0: One square. Step 1: Two squares, one atop the other. Step 2: Four squares in a two by two square. Step 3: Eight squares in a two by four rectangle.</p> — Pattern B

Write an equation to represent the number of small squares at Step in Pattern B.
Is the function linear, quadratic, or exponential?

Complete the table:

, step number	, number of small squares
0
1
2
3
4
5
6
7
8

How would the two patterns compare if they continue to grow? Make 1–2 observations.

Student Lesson Summary

The graphs of quadratic functions and the graphs of exponential functions with a base that is greater than 1 both curve upward. To compare the two, let’s look at the quadratic expression and the exponential expression .

A table of values shows that is initially greater than , but eventually becomes greater.


1	3	2
2	12	4
3	27	8
4	48	16
5	75	32
6	108	64
7	147	128
8	192	256

Here's why exponential growth eventually overtakes quadratic growth.

When increases by 1, the exponential expression always increases by a factor of 2.
The quadratic expression increases by different factors, depending on , but these factors get smaller. For example, when increases from 2 to 3, the factor is or 2.25. When increases from 6 to 7, the factor is or about 1.36. As increases to larger and larger values, grows by a factor that gets closer and closer to 1.

In general, quadratic functions change with a factor that gets closer and closer to 1 as the input to the function gets larger. Exponential functions always grow with the same factor, so if that growth factor is greater than 1, then the exponential function will eventually grow faster than any quadratic function will.

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Comparing Quadratic and Exponential Functions

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