How to find the exponential function given two points?
Understand the Problem
The question is asking for a method to determine the exponential function that passes through two specified points. This typically involves using the coordinates of the given points to set up equations based on the general form of an exponential function and then solving for the parameters to find the specific function.
Answer
The exponential function is $y = 2^x$.
Answer for screen readers
The exponential function that passes through the points $(1, 2)$ and $(3, 8)$ is given by:
$$ y = 2^x $$
Steps to Solve
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Identify the points Let’s say the two points are given as $(x_1, y_1)$ and $(x_2, y_2)$. For instance, assume these points are $(1, 2)$ and $(3, 8)$.
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Set up the exponential function The general form of an exponential function is given by: $$ y = ab^x $$ Where:
- $a$ is the initial value (when $x=0$),
- $b$ is the growth or decay factor.
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Create equations from the points Substituting the first point $(x_1, y_1)$ into the equation: $$ y_1 = ab^{x_1} $$ Substituting the second point $(x_2, y_2)$: $$ y_2 = ab^{x_2} $$
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Rewrite the equations From the substitutions, we have:
- Equation 1: $$ 2 = ab^1 $$
- Equation 2: $$ 8 = ab^3 $$
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Solve the first equation for $a$ From the first equation, isolate $a$: $$ a = \frac{2}{b} $$
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Substitute for $a$ in the second equation Now substitute $a$ into the second equation: $$ 8 = \left( \frac{2}{b} \right) b^3 $$
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Simplify and solve for $b$ Simplifying gives: $$ 8 = 2b^2 $$ To isolate $b^2$, divide both sides by 2: $$ b^2 = 4 $$ Taking the square root, we find: $$ b = 2 \text{ (since $b > 0$ for an exponential function)} $$
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Substitute back to find $a$ Substituting $b = 2$ back into the equation for $a$: $$ a = \frac{2}{2} = 1 $$
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Write the final exponential function Now we can write the exponential function: $$ y = 1 \cdot 2^x $$ or simply $$ y = 2^x $$
The exponential function that passes through the points $(1, 2)$ and $(3, 8)$ is given by:
$$ y = 2^x $$
More Information
This function indicates that the output doubles for every increase of 1 in the input value of $x$. The parameters $a$ and $b$ were found through the coordinates of the points provided.
Tips
- Forgetting that the base $b$ must be positive for an exponential function.
- Not substituting the correct values into equations or miscalculating while solving for $a$ and $b$.
- Assuming both $a$ and $b$ can be freely chosen without consideration of the provided points.
