How to find an exponential function from 2 points?
Understand the Problem
The question is asking how to determine an exponential function based on two specific points. To find such a function, we typically denote it in the form y = ab^x, where 'a' and 'b' are constants. Using the coordinates of the two points, we can create a system of equations to solve for 'a' and 'b'.
Answer
The exponential function will be in the form $y = ab^x$, with constants 'a' and 'b' determined by the given points.
Answer for screen readers
The final answer will be the specific exponential function in the form: $$ y = ab^x $$ where specific values for 'a' and 'b' are found based on the given points.
Steps to Solve
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Identify the points From the problem, note the two points given. Let's say they are $(x_1, y_1)$ and $(x_2, y_2)$.
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Set up the equations Using the general form of the exponential function $y = ab^x$, create a system of equations based on the two points:
- For the first point $(x_1, y_1)$: $$ y_1 = ab^{x_1} $$
- For the second point $(x_2, y_2)$: $$ y_2 = ab^{x_2} $$
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Solve for 'a' From the first equation, solve for 'a': $$ a = \frac{y_1}{b^{x_1}} $$
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Substitute 'a' in the second equation Substitute the expression for 'a' into the second equation: $$ y_2 = \left(\frac{y_1}{b^{x_1}}\right)b^{x_2} $$
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Rearrange to isolate 'b' Rearranging gives: $$ y_2 b^{x_1} = y_1 b^{x_2} $$
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Solve for 'b' Divide both sides by $b^{x_1}$: $$ y_2 = y_1 \cdot b^{x_2 - x_1} $$ Now solve for $b$: $$ b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}} $$
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Plug 'b' back into the equation for 'a' Now that you have 'b', substitute it back into the equation for 'a': $$ a = \frac{y_1}{b^{x_1}} $$
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Write the exponential function With both 'a' and 'b' determined, write the exponential function: $$ y = ab^x $$
The final answer will be the specific exponential function in the form: $$ y = ab^x $$ where specific values for 'a' and 'b' are found based on the given points.
More Information
This method of determining an exponential function is widely applicable in growth and decay scenarios, such as population growth, radioactive decay, and many others. By utilizing two points, we can uniquely determine the parameters for the exponential curve.
Tips
- Skipping the step of isolating 'a' and misunderstanding how to substitute it back into the second equation.
- Confusing the order of points which may lead to incorrect parameter values for 'a' and 'b'.
- Miscalculating exponents when solving for 'b'.
