The graph is f(×)=2.b^×+q determine the equation of f the graph is a exponential and it cuts on negative 1 the curve line going down right and the asymptote begin negative 6

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.

Explore how to solve the exponential function f(x) = 2 * b^x + q, given specific details of the y-intercept and asymptote. Learn the step-by-step process including finding the base 'b' and determining the function's behavior. Suitable for high school students learning about exponential functions.

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