Math Lesson 13.3.3 - Finding the Formula of an Unknown Exponential Function

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Welcome to our Math lesson on Finding the Formula of an Unknown Exponential Function, this is the third lesson of our suite of math lessons covering the topic of Modelling Curves using Logarithms, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Formula of an Unknown Exponential Function of the form y(x) = ba^x when the Graph is given

We saw in the above examples that in all cases, the graph of an exponential function is curved. This means we need to consider a large number of points in order to plot an accurate graph. However, there are some "tricks", which help identify the function's formula by considering only a few points from the graph. For this, we must apply the short procedure below.

Step 1: Find the y-intercept, as this point helps identify the value of the constant b. This is because the y-intercept has the horizontal coordinate x = 0. In this way, we have

For example, if the y-intercept of an exponential graph is y = 2, the function's general formula is

Step 2: Choose the coordinates of a known point of the graph to substitute in the general formula of the function in order to find the value of the base a. In this way, the formula of the unknown exponential function is already found.

For example, it is easy to notice in the graph below that the y-intercept is at y = -2.

Therefore, we have b = -2 and the function y(x) = b · a^x now becomes y(x) = -2 · a^x.

The next step involves the substitution of a known point from the graph in the formula. We can choose for example the point (1, -8), i.e. x = 1 and y = -8. In this way, we obtain for the base a:

Therefore, the function shown by the graph in the figure is

Example 1

Find the exponential functions indicated by the graphs below.

Solution 1

1. The y-intercept is y = 3. Therefore, b = 3. Now, let's check for any clear points on the graph. The point (1, -1.6) for example, is a clear point, where x = -8/5 and y = 1. Therefore, substituting the above values in the function
   y(x) = b ∙ a^x
   to obtain
   1 = 3 ∙ a^-8/5
   1/a - ^8/5 = 3/1
   a^-8/5 = 3
   (a^-8/5 )^5/8 = 3^5/8
   a = 3^5/8
   a = √3⁸
   = 2
   Therefore, the function shown on the graph has the formula
   y(x) = 3 ∙ 2^x
2. We use the same procedure as above. Thus, from the figure, it is easy to see that the y-intercept is y = -2. Therefore, we have b = -2. Again, from the figure, it is evident that point (1, -6) is a point of the graph. Hence, for x = 1 and y = -6 we obtain
   y(x) = b ∙ a^x
   -6 = -2 ∙ a¹
   a = -6/-2
   a = 3
   Therefore, the formula of this exponential function is
   y(x) = -2 ∙ 3^x

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