To find the equation of an exponential function based on two points, we can assume the function has the general form:

$y = ab^x$

where:

• $a$ is a constant (the initial value or vertical intercept),
• $b$ is the base (the growth/decay factor),
• $x$ is the exponent (independent variable),
• $y$ is the dependent variable (output).

Steps to Find the Exponential Equation:

1. Set up the equation for each point. If you have two points $(x_1, y_1)$ and $(x_2, y_2)$, substitute them into the general form.

   • For $(x_1, y_1)$: $y_1 = ab^{x_1}$
   • For $(x_2, y_2)$: $y_2 = ab^{x_2}$

2. Solve the system of equations. You now have two equations: $y_1 = ab^{x_1}$ $y_2 = ab^{x_2}$ To eliminate $a$, divide the second equation by the first: $\frac{y_2}{y_1} = \frac{ab^{x_2}}{ab^{x_1}} = b^{x_2 - x_1}$ This simplifies to: $b = \left( \frac{y_2}{y_1} \right)^{\frac{1}{x_2 - x_1}}$ Now you have the value of $b$, the base of the exponential function.

3. Find $a$. Substitute the value of $b$ back into one of the original equations to solve for $a$: $y_1 = ab^{x_1}$ Solving for $a$: $a = \frac{y_1}{b^{x_1}}$

4. Write the final equation. With both $a$ and $b$ known, the equation for the exponential function is: $y = ab^x$

Example:

Given the two points $(1, 2)$ and $(3, 18)$:

1. From $(1, 2)$: $2 = ab^1 \Rightarrow 2 = ab$
2. From $(3, 18)$: $18 = ab^3$
3. Divide the second equation by the first: $\frac{18}{2} = \frac{ab^3}{ab} \Rightarrow 9 = b^2$ Solving for $b$: $b = 3$
4. Substitute $b = 3$ into $2 = ab$: $2 = a \cdot 3 \Rightarrow a = \frac{2}{3}$
5. The final equation is: $y = \frac{2}{3} \cdot 3^x$

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Would you like more details on any of the steps? Here are five related questions to explore further:

1. How do you determine whether an exponential function represents growth or decay?
2. How can you use logarithms to solve for unknown exponents in exponential equations?
3. Can you fit an exponential model if the points are not exactly on a curve?
4. How do exponential functions compare to linear functions in terms of growth rates?
5. What are real-world examples where exponential models are useful?

Tip: Always check your two points with the derived function to ensure accuracy!