Math Lesson 15.5.2 - Exponential Function's Graph. Asymptotes of Exponential Graphs

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Welcome to our Math lesson on Exponential Function's Graph. Asymptotes of Exponential Graphs, this is the second lesson of our suite of math lessons covering the topic of Exponential Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Exponential Function's Graph. Asymptotes of Exponential Graphs

First, let's assume the base of all exponential functions discussed in this tutorial to be positive unless the existence of a negative base is stated explicitly. Moreover, we will first see how to plot the graph of the simplest form of an exponential function, i.e. of the function

Thus, since a positive number raised at any power is always positive, it is obvious that the y-values in the simplest form of an exponential function shown above are positive as well. Therefore, we have some limitations in the possible values an exponential function can take. Exponential functions always take the sign of the base a for every value of the independent variable. For example, in the function

all values of y are positive, for any value of the independent variable x chosen because the base a is positive (a = 2). Thus, for x = 3 we have y(3) = 23 = 8 (positive). Likewise, for x = -6, we have y(-6) = 2 - 6 = 1/26 = 1/64 (positive). Therefore, for this function, the Y-axis acts as a horizontal asymptote, as shown in the figure below.

On the other hand, the graph of an exponential function doesn't have any vertical asymptote, as there are no limitations in the values of x in such functions.

When the constant t is present in an exponential function, it makes the graph shift by t units above or below the same function that does not include this constant (above when t > 0 and below when t < 0). For example, the graph of the exponential function y(x) = 2x + 1 is one unit above the graph of y(x) = 2x because t = 1. In such cases, the function y(x) = ax is known as the parent function of y(x) = ax + t. This is because the second function derives from the first one with some extra accessories (here, by adding t to each y-value of the parent function). Hence, it is obvious that both functions have the same features but do not occupy the same position in the coordinates system. Look at the figure below where both graphs are shown in the same coordinates system.

Example 2

Without plotting the graph, find the equation of asymptotes in the following exponential equations and interpret the results.

1. y = 4x - 1/3
2. y = 3x + 2

Solution 2

First, let's find the asymptotes of the two parent functions, y = 4x and y = 3x. Thus, since both bases are positive (4 and 3 respectively), all y-values in the two functions are positive. Therefore, y = 0 is the horizontal asymptote for both parent functions. As for the given functions, they are obtained by adding or subtracting the value of the constant t to the corresponding parent functions. This constant also gives the horizontal asymptote of each functions. In this way, we have

1. a = 4 and t = -1/3. The horizontal asymptote (H.A.) is y = t = -1/3. In this way, the graph of y(x) = 4x - 1/3 is 1/3 units below the graph of y(x) = 4x.
2. a = 3 and t = 2. The horizontal asymptote (H.A.) is y = t = 2. In this way, the graph of y(x) = 3x + 2 is 2 units above the graph of y(x) = 3x.

You have reached the end of Math lesson 15.5.2 Exponential Function's Graph. Asymptotes of Exponential Graphs. There are 8 lessons in this physics tutorial covering Exponential Graphs, you can access all the lessons from this tutorial below.

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5. Check your calculations for Types of Graphs questions with our excellent Types of Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Types of Graphs Calculators by iCalculator™ below.
6. Continuing learning types of graphs - read our next math tutorial: Circle Graphs

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