What would the best exponential equation be for these points: 24, 274524 - 25, 276954 - 26, 292476

To find the best exponential equation for the given points, we need to model the data using an exponential function of the form:

\[
y = a \cdot e^{bx}
\]
where:
- \(y\) is the dependent variable (output),
- \(x\) is the independent variable (input),
- \(a\) and \(b\) are constants to be determined.

### Steps to find the equation:
1. Use the points \( (x_1, y_1) = (24, 274524) \), \( (x_2, y_2) = (25, 276954) \), and \( (x_3, y_3) = (26, 292476) \).
2. Fit these points to the general form \( y = a \cdot e^{bx} \), by solving for \(a\) and \(b\).
3. This can be done by transforming the equation into a linear form using natural logarithms and applying linear regression to determine \(a\) and \(b\).

I'll run through these calculations now to find the best fit exponential equation.

Give me a moment to calculate this.The best-fit exponential equation for the given points is:

\[
y = 125867.17 \cdot e^{0.03216x}
\]

Where:
- \( a = 125867.17 \)
- \( b = 0.03216 \)

This equation models the relationship between \(x\) and \(y\) based on the provided data.

Would you like more details on how this was derived, or do you have any further questions?

Here are 5 questions you might consider next:
1. How do logarithms help linearize an exponential equation?
2. What is the significance of the base \(e\) in the exponential equation?
3. How can you determine if the exponential model fits well to the data points?
4. Can you apply this model to predict future values (e.g., \(x = 27\))?
5. What is the interpretation of the constants \(a\) and \(b\) in the context of this model?

**Tip:** When fitting data to an exponential model, checking the residuals (the difference between observed and predicted values) can help assess the model's accuracy.

What would the best exponential equation be for these points: 24, 274524 - 25, 276954 - 26, 292476?

Discover how to find the best exponential equation for the given data points: 24, 274524 - 25, 276954 - 26, 292476. Using concepts like exponential functions, natural logarithms, and linear regression, this problem is ideal for Grades 11-12 students.

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