Variables As Exponents: Practice Problems

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

We see variables as exponents in a variety of problems in algebra. To better our understanding of this, we will look at some simple expressions and equations which contain variables as exponents. We will view these functions as equations algebraically, and as lines graphically.

Variables as Exponents

We are exposed to several types of functions in the study of algebra. For instance, we see functions as straight lines in the form of y = mx + b, and functions as curves with an exponent, e.g. power functions such as y = x2. In most cases, we are able to plot points and perhaps draw a graph of each type of function. Another type of function contains a variable as an exponent, and this is called an exponential function of the form y = ax, where a is some number (a constant) and x is a variable.

Equations with Exponents

It is very important to note that before we begin, we will want to make use of a calculator in this lesson. Furthermore, when solving for the following exponents with our calculator, we will want to make use of plenty of parentheses to ensure the proper order of operations.

Example # 1: y = 2x

Say we are given the equation of y = 2x. Can we sketch a graph of this function? We will input a variety of x-values into this equation and get corresponding y-values out. For instance, if we wish to find a point at x = 1, we plug x = 1 into our equation y = 2(1) = 2. We see that at x = 1, y = 2; therefore, we have a point at (1 , 2). We obtain more x- and y-values, or points, for y = 2x, from x = -3 to x = 3. Note that we should close negative exponents in parenthesis. For instance, when x = -1, y should equal 2(-1) from our calculator or y = .5. Another way of looking at the raising of a number to a negative exponent like y = 2(-1) is as 1 / 2(1).

Table of y = 2 x

Once we have these points as shown above, we plot them in the xy-plane and draw a line connecting them:

Grapf of y = 2 x

Notice that in this graph, y increases steadily with respect to x from x = -3 to x = 0 until x becomes greater than 0. At this point, y starts to increase exponentially, that is to say very rapidly.

Example # 2: y = -(2x)

For y = -(2x), we follow the same procedure as above. If we wish to find y when x = 1, we plug 1 into y = -(21) and we solve for y. We must treat this negative (-) in front of the 2 as (-1); therefore, (-1)(21) = -2. Furthermore, we should close any negative exponents in parentheses as well. For instance, if we had x = -1, we would enter the following for y into our calculator: y = (-1)(2(-1)), where we would find that y = -.5. We input values for x between x = -3 and x =3 to find corresponding y-values:

Table of y = -(2 x)

We plot these points and draw the graph:

Graph of y = -(2 x)

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