Finding The Exponential Function Formula Given Two Points

Hey guys! Today, we're diving into the fascinating world of exponential functions. Specifically, we're going to tackle the challenge of finding the formula for an exponential function that passes through two given points. This is a common problem in mathematics, and mastering it will give you a solid understanding of how exponential functions work. So, let's jump right in and break down the steps involved. We'll make sure everything is crystal clear, and by the end of this article, you'll be able to confidently solve similar problems. Let's get started!

Problem Statement

The problem we're tackling today is this: Find a formula for the exponential function passing through the points $\left(-3, \frac{3}{125}\right)$ and $(1,15)$. This means we need to determine the equation of an exponential function that fits these two specific points on a graph. Exponential functions have a unique form, and understanding this form is key to solving the problem. So, let's break it down step by step and make sure we understand the basics before diving into the calculations. First, let's understand the general form of an exponential function.

Understanding Exponential Functions

To begin, let's quickly recap what an exponential function actually is. An exponential function has the general form $y = ab^x$, where:

• y$ is the dependent variable.

• x$ is the independent variable.

• a$ is the initial value or the y-intercept (the value of $y$ when $x = 0$).

• b$ is the base, which is a positive constant not equal to 1. It determines the rate of growth or decay.

This form is crucial because it tells us how the function behaves. If $b > 1$, the function represents exponential growth, meaning the values of $y$ increase rapidly as $x$ increases. If $0 < b < 1$, the function represents exponential decay, where the values of $y$ decrease as $x$ increases. The constant $a$ simply scales the function vertically. Knowing this, our goal is to find the specific values of $a$ and $b$ that make the function pass through the given points.

Setting up the Equations

Now, armed with the general form, we can use the given points to set up a system of equations. This is a critical step because it transforms the problem into an algebraic one that we can solve systematically. We have two points: $\left(-3, \frac{3}{125}\right)$ and $(1,15)$. We'll plug these into our general form $y = ab^x$ to create two equations.

For the point $\left(-3, \frac{3}{125}\right)$, we have:

$$\frac{3}{125} = ab^{-3}$$

And for the point $(1,15)$, we have:

$$15 = ab^1$$

So now we have a system of two equations:

1. $$\frac{3}{125} = ab^{-3}$$

2. $$15 = ab$$

This system of equations is what we need to solve to find the values of $a$ and $b$. We can use various methods to solve it, such as substitution or elimination. In the next section, we'll walk through the substitution method, which is a straightforward approach for this type of problem. Stick with us, and you'll see how easily we can crack this!

Solving for $a$ and $b$

To solve for $a$ and $b$, we'll use the substitution method, which involves solving one equation for one variable and then substituting that expression into the other equation. This will leave us with a single equation in one variable, which we can solve. Let's start by solving the second equation, $15 = ab$, for $a$:

$$a = \frac{15}{b}$$

Now that we have an expression for $a$, we can substitute this into the first equation:

$$\frac{3}{125} = \left(\frac{15}{b}\right)b^{-3}$$

This substitution is a crucial step because it eliminates one variable, making the equation solvable. Now we have an equation only in terms of $b$, which we can simplify and solve. Let's simplify the equation by multiplying both sides by $b$:

$$\frac{3}{125} = 15b^{-4}$$

Next, we'll isolate $b^{-4}$ by dividing both sides by 15:

$$b^{-4} = \frac{3}{125 \cdot 15} = \frac{3}{1875} = \frac{1}{625}$$

Now we need to solve for $b$. Since $b^{-4} = \frac{1}{625}$, we can take the reciprocal of both sides to get:

$$b^4 = 625$$

To find $b$, we take the fourth root of both sides:

$$b = \sqrt[4]{625} = 5$$

So, we've found that $b = 5$. Now that we have $b$, we can easily find $a$ by plugging $b$ back into our expression for $a$:

$$a = \frac{15}{b} = \frac{15}{5} = 3$$

Therefore, $a = 3$. We now have both $a$ and $b$, which are the key components of our exponential function. With these values in hand, we can write the final formula for the exponential function.

Constructing the Formula

Now that we've found the values of $a$ and $b$, we can construct the formula for the exponential function. Remember the general form of an exponential function is $y = ab^x$. We found that $a = 3$ and $b = 5$, so we simply plug these values into the general form:

$$y = 3 \cdot 5^x$$

This is the specific exponential function that passes through the points $\left(-3, \frac{3}{125}\right)$ and $(1,15)$. To ensure our solution is correct, it's always a good idea to verify it by plugging in the given points and checking if they satisfy the equation.

Verifying the Solution

To verify that our formula $y = 3 \cdot 5^x$ is correct, we'll plug in the given points and see if they satisfy the equation. This step is crucial to ensure we haven't made any mistakes in our calculations. Let's start with the point $\left(-3, \frac{3}{125}\right)$. Plugging in $x = -3$, we get:

$$y = 3 \cdot 5^{-3} = 3 \cdot \frac{1}{5^3} = 3 \cdot \frac{1}{125} = \frac{3}{125}$$

This matches the $y$-coordinate of the point, so the equation holds true for this point. Now let's check the second point, $(1,15)$. Plugging in $x = 1$, we get:

$$y = 3 \cdot 5^1 = 3 \cdot 5 = 15$$

This also matches the $y$-coordinate of the point, so the equation holds true for this point as well. Since the equation holds true for both points, we can confidently say that our solution is correct. We've successfully found the exponential function that passes through the given points!

Final Answer

Therefore, the formula for the exponential function passing through the points $\left(-3, \frac{3}{125}\right)$ and $(1,15)$ is:

$$y = 3 \cdot 5^x$$

This is our final answer. We've walked through the entire process, from understanding the general form of an exponential function to setting up and solving equations, and finally verifying our solution. Hopefully, this breakdown has made the process clear and understandable. Remember, practice makes perfect, so try tackling similar problems to solidify your understanding. Keep up the great work, guys!

Conclusion

In this comprehensive guide, we've successfully navigated the process of finding the formula for an exponential function that passes through two given points. We started by understanding the general form of an exponential function, $y = ab^x$, and identified the key components: the initial value $a$ and the base $b$. We then translated the problem into a system of equations by plugging in the coordinates of the given points. The substitution method proved to be a powerful tool in solving this system, allowing us to find the values of $a$ and $b$ efficiently. Once we had these values, constructing the formula was as simple as plugging them back into the general form.

Verifying our solution was a critical step, ensuring that our calculations were accurate and that the resulting function indeed passed through the given points. This step reinforced our confidence in the final answer. By breaking down the problem into manageable steps and providing clear explanations, we've aimed to make this process accessible and understandable. Remember, the key to mastering these concepts is practice. Try applying these techniques to similar problems, and you'll find your skills and understanding growing stronger. Whether you're a student tackling algebra or simply someone interested in mathematical problem-solving, we hope this guide has been a valuable resource. Keep exploring the fascinating world of mathematics, and don't hesitate to tackle new challenges. You've got this!