What is the multiplicative rate of change for the exponential function graphed to the left?

The multiplicative rate of change for an exponential function is the base of the exponential expression in the function's formula. The general form of an exponential function is:

\[ f(x) = a \cdot b^x \]

where:
- \( a \) is the initial value (the function's value when \( x = 0 \)),
- \( b \) is the base, representing the multiplicative rate of change or the growth/decay factor,
- \( x \) is the exponent.

If you are looking at a graph and trying to determine the multiplicative rate of change (i.e., the value of \( b \)), you would typically need to identify two key points on the curve:

1. The y-intercept, which is the value of the function when \( x = 0 \).
2. Another point on the graph to determine the rate of increase or decrease.

### Steps to Determine the Rate of Change from a Graph:

1. **Identify the y-intercept**: This gives you the initial value, \( a \).
2. **Choose another point on the graph**: Let this point be \((x_1, y_1)\).
3. **Use the function form**: With these two points, you can set up the equation:
   \[ y_1 = a \cdot b^{x_1} \]

4. **Solve for \( b \)**:
   \[ b = \left(\frac{y_1}{a}\right)^{1/x_1} \]

Since I can't view the graph you're referencing, you would need to use these steps to identify the specific values of \( a \) and \( b \) from the graph. The value of \( b \) will give you the multiplicative rate of change of the exponential function.

If you have the specific points or the function formula, I can help you calculate \( b \) directly. Let me know if you have more information or need further assistance!

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Do you want more details or have any questions? Here are some related questions that might interest you:

1. How do you find the y-intercept of an exponential function?
2. What does the base \( b \) indicate in terms of growth and decay?
3. How can you determine if an exponential function represents growth or decay?
4. What are the properties of exponential functions?
5. How does the initial value \( a \) affect the graph of an exponential function?
6. How do transformations affect the graph of an exponential function?
7. What are common real-life applications of exponential functions?
8. How do exponential functions compare to linear functions?

**Tip:** In an exponential growth function, if \( b > 1 \), the function represents growth, while if \( 0 < b < 1 \), it represents decay.

Learn how to calculate the multiplicative rate of change for an exponential function graph. Understand the exponential function formula f(x) = a * b^x and the rate of change formula b = (y1 / a)^(1 / x1). This problem involves interpreting graphs and identifying key points to determine the rate of change. Suitable for students in Grades 9-12.

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