To write the exponential function, we use the general formula:

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

where:

• $a$ is the initial value,
• $r$ is the rate of change (expressed as a decimal),
• $x$ is the input variable (time or another independent variable).

For decay rates, $r$ will be negative, so the formula becomes:
$f(x) = a(1 - r)^x$

Here are the solutions for the given values:

1. Initial Value = 180, Growth Rate = 12%
   $f(x) = 180(1 + 0.12)^x = 180(1.12)^x$

2. Initial Value = 56, Growth Rate = 7%
   $f(x) = 56(1 + 0.07)^x = 56(1.07)^x$

3. Initial Value = 104, Growth Rate = 6.5%
   $f(x) = 104(1 + 0.065)^x = 104(1.065)^x$

4. Initial Value = 73, Growth Rate = 103%
   $f(x) = 73(1 + 1.03)^x = 73(2.03)^x$

5. Initial Value = 176, Decay Rate = 29%
   $f(x) = 176(1 - 0.29)^x = 176(0.71)^x$

6. Initial Value = 122, Decay Rate = 9%
   $f(x) = 122(1 - 0.09)^x = 122(0.91)^x$

7. Initial Value = 197, Decay Rate = 0.1%
   $f(x) = 197(1 - 0.001)^x = 197(0.999)^x$

Let me know if you’d like further clarification or a detailed explanation of how these formulas are derived!

Related Questions:

1. What happens to the exponential function as $x \to \infty$ for growth and decay rates?
2. How do you convert a percentage growth rate to a decimal form?
3. What is the difference between linear and exponential growth?
4. How would a negative initial value affect the exponential function?
5. Can exponential functions model real-life situations? Provide examples.

Tip: Always ensure the rate is correctly expressed as a decimal (e.g., 12% = 0.12) when substituting into the formula