Given the information provided, we can determine the equation of the exponential function $f(x) = 2 \cdot b^x + q$.

Key Details:

1. Form of the Function: $f(x) = 2 \cdot b^x + q$

   • $2$ is the coefficient (amplitude).
   • $b$ is the base of the exponential function.
   • $q$ is the vertical shift, representing the horizontal asymptote.

2. Y-Intercept: The graph cuts the y-axis at $-1$. This means when $x = 0$, $f(0) = -1$.

3. Asymptote: The horizontal asymptote is at $y = -6$, meaning $q = -6$.

Step-by-Step Solution:

1. Substitute the known asymptote: $f(x) = 2 \cdot b^x - 6$ Here, $q = -6$.

2. Use the y-intercept to find $b$: Since the graph cuts the y-axis at $-1$, substitute $x = 0$ and $f(0) = -1$ into the equation: $-1 = 2 \cdot b^0 - 6$ $-1 = 2 \cdot 1 - 6$ $-1 = 2 - 6$ $5 = 2b^0$

   Now, solve for $b$: $2 = 1.3 b$

   Which simplifies to: $b = 0.833$

Final Equation:

So, the equation of the function $f(x)$ is approximately: $f(x) = 2(0.833^x) - 6$

This equation describes an exponential decay function that decreases as $x$ increases.

Would you like any further details or have any questions? Here are a few related questions you might explore:

1. How do transformations affect the graph of an exponential function?
2. How would the graph look if the base $b$ were greater than 1?
3. How can you determine the equation of an exponential function from a graph with a different asymptote?
4. What happens to the graph if the coefficient 2 is replaced by a different number?
5. Can you explain how to graph exponential functions without using a calculator?

Tip: When graphing, always check the y-intercept and asymptote first. They provide critical information about the function's behavior.