Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=2+e^{3 x}$$

Step-by-Step Solution

Answer

The graph of \(f(x)=2+e^{3 x}\) is an exponential growth curve. As \(x\) approaches negative infinity, the function gets closer to the line \(y=2\) but never touches it. Hence, the horizontal asymptote of the function is \(y=2\).

Step 1: Constructing a Table of Values

Using a graphing tool, input the function \(f(x)=2+e^{3 x}\). Generate a range of values from x to f(x). For example, find the values of f(x) for x=-1, x=0, x=1, and x=2.

Step 2: Sketching the Graph

Now, using the table of values generated in Step 1, plot each point on the graph. As you move forward, you'll begin to notice the general shape of the graph, which should resemble an exponential growth curve.

Step 3: Identifying Asymptotes

Observe the graph as x approaches negative infinity. The function will get closer and closer to the horizontal line y=2 but never actually touch it. This line is the horizontal asymptote for the function \(f(x)=2+e^{3 x}\), and it can be written as y=2.

Key Concepts

Table of Values

Understanding the table of values is crucial when graphing functions, especially exponential functions like f(x) = 2 + e^{3x}. By selecting input values for x, you can calculate the corresponding f(x) values. It's practical to choose a range that includes both negative and positive values of x, to capture the growth behavior of the exponential function.

For instance, if you choose x values such as -2, -1, 0, 1, and 2, you would compute the respective f(x) using the given expression. Remember that even slight changes in x can result in large variations in f(x) due to the nature of exponential growth. This table helps us to visualize and prepare the data needed to plot the graph accurately.

Sketching Graphs

Once you have your table of values, sketching the graph becomes an interpretative task. Begin by plotting each (x, f(x)) coordinate on the graph. With exponential functions such as f(x) = 2 + e^{3x}, expect a rapid increase as x moves from negative to positive.

In sketching the curve, it's essential to note the smooth and continuous nature of exponential functions. The plotted points guide you in drawing a curve that reflects exponential growth: starting off near the asymptote, increasing slowly at first, and then accelerating upwards as x increases.

Identifying Asymptotes

An asymptote is a line that a graph approaches but never touches or crosses. For the exponential function f(x) = 2 + e^{3x}, as x approaches negative infinity, the e^{3x} term approaches zero, making the function value come closer and closer to 2. Thus, the horizontal line y = 2 is a horizontal asymptote of the function.

This asymptote acts as a boundary for the graph. As you graph the function, make sure the curve approaches but does not cross or touch the line y = 2 as x decreases. Visualizing the asymptote helps with understanding the behavior of the function at extreme values of x and ensures the accuracy of the graph.