Question

1. your company’s sales of $13,000 increase by 15% each year. write the function and fill in a table of values that represents this situation.

t	-2	-1	0	1	2	3

$f(t) = 13000(1.15)^t$

Explanation:

Step1: Define exponential growth function

The general form of an exponential growth function is $f(t) = P(1+r)^t$, where $P$ is the initial amount, $r$ is the growth rate, and $t$ is time. Here, $P = 13000$, $r = 0.15$, so:
$$f(t) = 13000(1+0.15)^t = 13000(1.15)^t$$

Step2: Calculate $f(-2)$ (2 years prior)

Substitute $t=-2$ into the function:
$$f(-2) = 13000(1.15)^{-2} = 13000 \times \frac{1}{(1.15)^2} = 13000 \times \frac{1}{1.3225} \approx 9829.86$$

Step3: Calculate $f(-1)$ (1 year prior)

Substitute $t=-1$ into the function:
$$f(-1) = 13000(1.15)^{-1} = 13000 \times \frac{1}{1.15} \approx 11304.35$$

Step4: Calculate $f(0)$ (initial year)

Substitute $t=0$ into the function (any number to 0 power is 1):
$$f(0) = 13000(1.15)^0 = 13000 \times 1 = 13000$$

Step5: Calculate $f(1)$ (1 year later)

Substitute $t=1$ into the function:
$$f(1) = 13000(1.15)^1 = 13000 \times 1.15 = 14950$$

Step6: Calculate $f(2)$ (2 years later)

Substitute $t=2$ into the function:
$$f(2) = 13000(1.15)^2 = 13000 \times 1.3225 = 17192.50$$

Step7: Calculate $f(3)$ (3 years later)

Substitute $t=3$ into the function:
$$f(3) = 13000(1.15)^3 = 13000 \times 1.520875 = 19771.38$$

Answer:

Function: $\boldsymbol{f(t) = 13000(1.15)^t}$

Table of values:

$t$	-2	-1	0	1	2	3