How to find an exponential equation from two points?
Understand the Problem
The question is asking for the method to derive an exponential equation given two specific points on a coordinate plane. It involves finding parameters (like the base and exponent) that will form an equation in the form y = ab^x, where 'a' and 'b' are constants derived from the provided points.
Answer
The exponential equation derived from the points is $y = ab^x$.
Answer for screen readers
The final exponential equation in the form $y = ab^x$ can be written once you’ve calculated the values of constants $a$ and $b$ based on the two points.
Steps to Solve
- Identify the Given Points
Let the two points be $(x_1, y_1)$ and $(x_2, y_2)$. Assign these values based on the specific coordinates you have.
- Set Up the System of Equations
Using the points and the exponential equation form $y = ab^x$, create two equations:
For the first point: $$ y_1 = ab^{x_1} $$
For the second point: $$ y_2 = ab^{x_2} $$
- Divide the Equations
To eliminate $a$, divide the second equation by the first: $$ \frac{y_2}{y_1} = \frac{ab^{x_2}}{ab^{x_1}} $$ This simplifies to: $$ \frac{y_2}{y_1} = b^{x_2 - x_1} $$
- Solve for 'b'
Next, take the $x_2 - x_1$ root of both sides to solve for $b$: $$ b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}} $$
- Find 'a'
Now substitute the value of $b$ back into one of the original equations to solve for $a$. For example, using the first point: $$ a = \frac{y_1}{b^{x_1}} $$
- Write the Final Exponential Equation
Substituting the values of $a$ and $b$ obtained into the equation $y = ab^x$ gives you the final exponential equation.
The final exponential equation in the form $y = ab^x$ can be written once you’ve calculated the values of constants $a$ and $b$ based on the two points.
More Information
Exponential equations are often used in various fields including biology for population growth, finance for compound interest, and physics for radioactive decay. Understanding how to derive these equations from points is fundamental in modeling real-world scenarios.
Tips
- Forgetting to take the root of the ratio correctly when finding $b$ can lead to incorrect results.
- Not substituting $b$ back into the equation properly when solving for $a$ can also cause errors.
- Mixing up the points can lead to inconsistent equations.
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