Exponential form3/23/2023 ![]() ![]() Substituting into our equation for a, we get: Substituting into the second equation gives us: Solving the first equation for a gives us: To solve this system, we will solve for a in the first equation and substitute into the second. So, our system of two equations in two unknowns is: Now, using the second point (3, 40), we substitute x = 3 and y = 40 to get: Using the first point (1, 10), we substitute x = 1 and y = 10 to get: ![]() Let’s say that you are given the points (1, 10) and (3, 40), which lie on the graph of an exponential function. Example 2: How To Find The Formula Of An Exponential Function With Two Points The graph of the exponential function y = 3 x. You can see the graph of this function below, which includes the two points (0, 1) and (2, 9). So, the exponential function in this case is y = 3 x or f(x) = 3 x. Using both a = 1 and b = 3 in the general formula for an exponential function, we get: Now, using the second point (2, 9), we substitute x = 2 and y = 9 to get: In this case, the exponent of 0 on b causes it to cancel out, which gives us a = 1. Using the first point (0, 1), we substitute x = 0 and y = 1 to get: We know that the general formula for an exponential function is given by: Let’s say that you are given the points (0, 1) and (2, 9), which lie on the graph of an exponential function. Example 1: How To Find The Formula Of An Exponential Function With Two Points Here are some examples to show how it works. Solve the system of two equations that you got from steps 1 & 2. Plug in the second point into the formula y = ab x to get your second equation. Plug in the first point into the formula y = ab x to get your first equation. Remember, there are three basic steps to find the formula of an exponential function with two points: How To Find The Formula Of An Exponential Function With Two Points Let’s take a closer look at each method and some examples. It all comes down to finding two points on the curve and using them to write two equations, then solving the system. If you are given the graph of an exponential function, you can simply choose two points on the curve and use the same method outlined above. If you are given a table of values for an exponential function, you can simply choose two points and use the same method outlined above. How To Find The Formula Of An Exponential Function We can then solve the system of equations. Once we have those two points, we can substitute each pair into the general formula for an exponential function:Īfter we substitute both points, we get two equations in two unknowns (the parameters a and b). To find the formula of an exponential function, all we need is two different points on the curve. (You can also watch a video summary of this article on YouTube). We’ll also dive into a few examples to make the concepts clear. In this article, we’ll take a look at the ways you can find the formula of an exponential function. ![]() Of course, we cannot find the formula of an exponential function from one point, since one point (one equation) does not give us enough information to solve for two parameters (the coefficient a and the base b). Given a graph or a table of values, we just need to choose two points and use the same method described above. ![]() So, how do you find the formula of an exponential function? We can find the formula of an exponential function by using two points on the curve, substituting them into the formula y = ab x, and solving the system of two equations in two unknowns. It is one thing to make a table or graph given the formula for an exponential function, but working backwards is another thing. Exponential functions come up often in algebra and calculus. ![]()
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