To analyze and help solve the problem shown in your image:

We are determining the values of $b$ and $k$ in the exponential function $f(x) = 3b^x + k$.

Observations from the table:

1. Horizontal Asymptote: $y = -2$ implies $k = -2$.
2. The given constant percent rate of change is 0.50. This indicates the growth/decay factor $b$.

To find $b$, observe that the percent rate of change corresponds to $b - 1$. Since the rate of change is 0.50 (50%), we can write: $b - 1 = -0.50 \quad \Rightarrow \quad b = 0.5$

Final answers:

• $b = 0.5$
• $k = -2$

The function:

$f(x) = 3(0.5)^x - 2$

Would you like further explanation on any of these steps?

Related Questions:

1. How do you determine the horizontal asymptote from an exponential function?
2. What is the significance of the constant percent rate of change?
3. How do you identify the $y$-intercept from an exponential function's equation?
4. What is the difference between growth and decay factors in exponential equations?
5. How can you verify the accuracy of the values of $b$ and $k$?

Tip: Remember that for exponential decay, $b$ is always between 0 and 1!