Exploring Linear, Exponential, and Quadratic Growth

7.4.2 Exploring Linear, Exponential, and Quadratic Growth

7.4.2 • Exploring Linear, Exponential, and Quadratic Growth

In this activity, you will compare two different patterns. In each pattern, the number of small squares is a function of the step number, $n$ .

1. In Pattern A, the length and width of the rectangle grow by 1 small square from each step to the next.

Pattern A

A pattern of squares. Step 0 has zero squares. Step 1 has 1 square. Step 2 has 2 rows with 2 squares each. Step 3 has 3 rows with 3 squares each.

a. Write an equation to represent the number of small squares at Step $n$ in Pattern A.

Compare your answer:

$f (n) = n^{2}$

b. Is the function in Pattern A linear, quadratic, or exponential?

It is quadratic.

c. Complete the table for Pattern A.

Step Number $n$	Number of Small Squares $f (n)$
$0$
$1$
$2$
$3$
$4$
$5$
$6$
$7$
$8$

Compare your answers:

Step Number $n$	Number of Small Squares $f (n)$
$0$	$0$
$1$	$1$
$2$	$4$
$3$	$9$
$4$	$16$
$5$	$25$
$6$	$36$
$7$	$49$
$8$	$64$

2. In Pattern B, the number of small squares doubles from each step to the next.

Pattern B

A pattern of squares. Step 0 has 1 column with 1 square in it. Step 1 has 1 column with 2 squares in it. Step 2 has 2 columns with 2 squares in each of them. Step 3 has 4 columns with 2 squares in each of them.

a. Write an equation to represent the number of small squares at Step $n$ in Pattern B.

Compare your answer:

$g (n) = 2^{n}$

b. Is the function in Pattern B linear, quadratic, or exponential?

It is exponential.

c. Complete the table for Pattern B.

Step Number $n$	Number of Small Squares $g (n)$
$0$
$1$
$2$
$3$
$4$
$5$
$6$
$7$
$8$

Compare your answers:

Step Number $n$	Number of Small Squares $g (n)$
$0$	$1$
$1$	$2$
$2$	$4$
$3$	$8$
$4$	$16$
$5$	$32$
$6$	$64$
$7$	$128$
$8$	$256$

3. How would the two patterns compare if they continue to grow? Make one to two observations.

Compare your answer:

The number of small squares in Pattern B grows much more quickly than the number of squares in Pattern A once $n$ is greater than $5$ . The patterns have the same number of squares when $n = 2$ and when $n = 4$ .

Self Check

Which of the following functions has a constant growth factor?

$f(x)=x+4$
$f(x)=4^x$
$f(x)=4x^2$
$f(x)=4x$

Additional Resources

Determining Growth Factors

Is the pattern below represented by a function that is linear, exponential, or quadratic?

A pattern of squares. Step 0 has 1 square. Step 1 has a t-shaped pattern that consists of three squares in two rows. The squares are arranged so that the square in the bottom row is offset from the two squares in the row above it so they form the shape of the letter t. Step 2 is a combination of three of these t-shapes in one row. So, there are 6 squares in row 2 and 3 squares on the bottom row off set from the adjoining squares above them. Step 3 consists of nine of these t-shapes. There are three t-shapes on the bottom and then six t-shapes in the next level above. To breakdown all of the rows, step 3 has a total of 4 rows. The bottom row has three squares with the row above it containing six squares. This set constitutes the first set of t-shapes. Then, row 3 contains 6 squares with row 4 containing 12 squares. This is the second set of t-shapes.

Step 1 - Draw a table. In the right column, determine the growth factor.

Step Number $x$	Number of Squares $f (x)$	Growth Factor
$0$	$1$	$3$
$1$	$3$	$3$
$2$	$9$	$3$
$3$	$27$	$3$

Step 2 - Write the function to represent the pattern.

$f (x) = 3^{x}$

Step 3 - Determine what type of function this is.

The function is exponential.

Try it

Determining Growth Factors

Is the pattern below represented by a function that is linear, exponential, or quadratic?

A pattern of squares. Step 0 has 0 squares. Step 1 has 3 columns with 1 square in each. Step 2 has 6 columns with 2 squares in each. Step 3 has 9 columns with 3 squares each.

Here is how to use the growth factor to determine if this pattern is represented by a linear, quadratic, or exponential function:

Step 1 - Create a table and determine the growth factor.

Step Number $x$	Number of Squares $f (x)$	Growth Factor
$0$	$0$
$1$	$3$
$2$	$12$	$4$
$3$	$27$	$2.25$

Step 2 - Determine the function.

$f (x) = 3 x^{2}$

Step 3 - Determine what type of function this is.

This is a quadratic function. The growth rate changes between each step.