Comparing Linear and Exponential Growth

5.15.4 Comparing Linear and Exponential Growth

5.15.4 • Comparing Linear and Exponential Growth

Cool Down

The linear function $g$ is defined by $g (x) = 4 x$ .

a. Show that $g (3)$ and $g (2)$ have a difference of 4.

b. Show that when the input value increases from $x$ to $x + 1$ , the output values $g (x + 1)$ and $g (x)$ have a difference of 4.

Compare your answer:

$g (3) = 4 \cdot 3 = 12$ , and $g (2) = 4 \cdot 2 = 8$ . The difference of 12 and 8 is 4.
$g (3) - g (2) = 4 \cdot (3) - 4 \cdot (2) = 4 \cdot (3 - 2) = 4$

b. $g (x + 1) - g (x) = 4 \cdot (x + 1) - 4 x = 4 \cdot (x + 1 - x) = 4 \cdot (1)$ , which equals 4.

The exponential function $h$ is defined by $h (x) = 4^{x}$ .

a. Show that $h (3)$ and $h (2)$ have a quotient of 4.

b. Show that when the input value increases from $x$ to $x + 1$ , the output values $h (x + 1)$ and $h (x)$ have a quotient of 4.

Compare your answer:

$h (3) = 4^{3} = 64$ , and $h (2) = 4^{2} = 16$ . The quotient is $\frac{64}{16}$ , which is 4.
$\frac{h (3)}{h (2)} = \frac{4^{3}}{4^{2}} = 4^{3 - 2} = 4$

b. $\frac{h (x + 1)}{h (x)} = \frac{4^{x + 1}}{4^{x}} = 4^{x + 1 - x} = 4^{1}$ , which equals 4.

Why Should I Care?

A hand holds a tablet displaying stacks of gold coins increasing in height, with the text "Power of Compounding" and example calculations showing how $1,000 grows to $1,210 with repeated 10% interest.

Exponential models can help you understand the power of investment. After two years, this investor made $210 off of an initial investment of $1000 because of the 10% interest rate.

Can you calculate how much money the investor will have made after twenty years?