To solve the problems, we need to write the exponential functions for each case based on the initial value and the rate provided. The general form of an exponential function is:

$f(x) = a(1 + r)^x$

where:

• $a$ is the initial value,
• $r$ is the growth rate (or decay rate, where decay rates are negative),
• $x$ is the independent variable.

Calculations:

1. 117, Growth Rate = 27% $f(x) = 117(1 + 0.27)^x = 117(1.27)^x$

2. 189, Growth Rate = 7% $f(x) = 189(1 + 0.07)^x = 189(1.07)^x$

3. 93, Growth Rate = 8.5% $f(x) = 93(1 + 0.085)^x = 93(1.085)^x$

4. 187, Growth Rate = 120% $f(x) = 187(1 + 1.20)^x = 187(2.20)^x$

5. 195, Decay Rate = 24% $f(x) = 195(1 - 0.24)^x = 195(0.76)^x$

6. 182, Decay Rate = 6% $f(x) = 182(1 - 0.06)^x = 182(0.94)^x$

7. 53, Decay Rate = 0.4% $f(x) = 53(1 - 0.004)^x = 53(0.996)^x$

Let me know if you need any further details or explanations about this!

Follow-up Questions:

1. Why do growth rates add to 1 and decay rates subtract from 1 in the formula?
2. What does the variable $x$ represent in these functions?
3. How does the rate impact the steepness of the graph?
4. What happens to the function values when the rate of decay is very small?
5. How do you interpret $f(0)$ in all these cases?

Tip:

Always convert percentage rates to decimals before using them in equations! For example, 27% becomes $0.27$.