Identifying an Exponential Equation Based on a Set of Values or Points:

One of the ways that you could be tested on your knowledge of exponential equations is by presenting you with a table of values and asking you to select which answer choice equation best corresponds to the data in the table:

There are a few ways to approach this using Desmos. First, we present a simple approach: for each answer choice, we enter the expression on the right side of the equation (you could enter the complete equations, but it's unnecessary), adding a slider for t, and we then set the value of t to 0, 1, and 2 using the slider or by typing in those values directly while comparing the resulting value of each expression to the corresponding entry in the Total Amount column in the table to see which expression produces matching values for t = 0, 1, and 2. Note that the slider for t is the second entry; that's where it's placed if you accept Desmos's offer to create a slider for t after you enter the first expression, so we left it there.

As it happens, in this problem you don't actually need to set t to 0, 1, and 2 to be able to select the correct answer choice because the starting value of t will be 1 when you accept Desmos's offer to create a slider for t, and each expression produces a different value when t = 1; only one of those values corresponds to the value in the Total Amount column for t = 1. In other problem setups, you might have to step through more values to determine which answer choice is correct.

Note that the value of the correct answer choice's expression when t = 1 isn't exactly the same as the value in the table, but the problem is careful to ask for which equation best represents the relationship expressed in the table.

Answer: C

Next, we present a rather sophisticated approach, but it's not out of reach for most students, and it's actually quite efficient. Rather than testing all the answer choices to see which equation matches the data, we'll give Desmos the data and let it figure out the equation for us. To accomplish this, we'll first create a table with two rows from the given data (you only need two pairs of function inputs/outputs, or, equivalently, two points, to determine an exponential function), and then we'll use the regression feature to ask Desmos to compute the components of the exponential equation that is represented by the data. How do we know it's an exponential equation? Two ways: all the answer choice are exponential equations, and the question itself states, "the table shows the exponential relationship."

You add a table item in Desmos by clicking the plus sign and selecting Table from the dropdown or by using the smarty's shortcut and typing "table" into an entry. You can stick with the default column headers, or you can set them to match the problem, in which case the first column would have the header t for the time in years and the second column would have the header n for the total amount in euros.

The form of an exponential equation in this scenario is n = a(1+r)^t, and we want to determine the values of a and r so we can pick the matching answer choice. To use regression to have Desmos determine those components of the exponential equation, we set up a regression statement. That looks just like an equation, except that instead of using an equals sign, we use the regression symbol ~. Desmos will use the values for n and t in the table to compute the values of a and r. You can then use those values to find the answer choice with the matching values (they might not match exactly, but that's ok—note that the problem asks for the equation that best represents the relationship).

Here's a variation where you're supplied with the data in the form of points on the graph of the function:

We can use Desmos's regression operator ~ to find the values of the constants a and b that are needed to construct an exponential function that passes through the given points, and we can then have Desmos evaluate the term ab.

Answer: C

Some problems involving exponential functions can be approached in several ways when using Desmos:

One straightforward approach here is to simply increase x by 1 and decrease f(0) by 80% to find f(1), then repeat that process to find f(2). Note that to decrease a value by 80%, we take 20% (100%-80%) of it, and also note the use of Desmos's percentage feature.

Answer: 3.44 or 86/25

An alternative approach is to construct the exponential function f, then evaluate f(2). The problem supplies the starting value of 86 and the change rate of 100% - 80% = 20%, which is represented as its decimal equivalent, 0.2, in the function.