Problem 41 - 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 asymptotes of the graph. $$f(x)=3^{x+2}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 3^{x+2}\) increases exponentially from left to right. It does not have any vertical asymptotes but has a horizontal asymptote at y=0.

Step 1: Construct a Table of Values

Using a graphing utility, input the function \(f(x) = 3^{x+2}\). Choose a range of x-values and calculate their corresponding f(x) values. These values will be used to draw the graph.

Step 2: Sketch the Graph

Once you have your table of values, plot these on the graph. The x-values are on the horizontal axis and the f(x) values are on the vertical axis. As you plot these points, observe the trend they create and extend this trend to create a smooth curve representing the function \(f(x) = 3^{x+2}\).

Step 3: Identify The Asymptotes

An asymptote is a line that a graph approaches but never touches. This graph doesn’t have any vertical asymptotes because the exponential function is defined for all real numbers. However, it does have a horizontal asymptote, which is the x-axis or the line y=0. This is because as x tends towards negative infinity, \(f(x) = 3^{x+2}\) approaches 0.

Key Concepts

Asymptotes of Graphs

In the realm of graphing functions, an asymptote is like a boundary that the graph approaches but never crosses or reaches. With exponential functions such as f(x) = 3^{x+2}, understanding asymptotes is crucial for accurately depicting the behavior of the graph.

Exponential graphs typically have a horizontal asymptote. For f(x) = 3^{x+2}, the horizontal asymptote is the line y = 0, which is the x-axis. This is because, as the x value goes to negative infinity, the f(x) value gets closer and closer to 0 but never actually reaches it. In other words, the graph flattens out as it extends leftwards but stays just above the x-axis. Unlike polynomial functions, which can have both horizontal and vertical asymptotes, the basic form of an exponential function will not have vertical asymptotes since they do not have undefined values for any real x.

When working with more complicated functions that combine exponential and other forms, such as rational expressions, you may encounter both horizontal and vertical asymptotes. Identifying these correctly is essential to sketching an accurate graph of the function.

Table of Values

Constructing a table of values is a stepping stone to visualizing a function when graphing it. It helps us get a sense of what the function looks like by providing specific points through which the graph will pass. For the exponential function f(x) = 3^{x+2}, you would choose values for x (both positive and negative) and calculate the corresponding values of f(x).

A good table of values should have a range broad enough to show the variation in the function's behavior, typically including values where the function increases or decreases sharply. In the case of our function, as x increases, f(x) will grow rapidly due to the exponential nature of the graph. As x decreases, f(x) will approach the asymptote. Remember, creating a comprehensive table of values aids in constructing a more accurate graph, as it includes information that represents the function's overall behavior.

Sketching Graphs

The act of sketching the graph of a function is a bridge between abstract mathematical concepts and visual understanding. After assembling a table of values, the points (derived from the table) are plotted on the Cartesian plane to lay the groundwork for the graph. For the function f(x) = 3^{x+2}, as you plot the points, you will start to notice a distinct pattern or shape forming, specifically, that of an exponential increase.

Graphing software may help visualize the sketch, but the principles remain the same when drawn by hand. Start by marking the points from the table on your graph, ensuring your x-axis and y-axis are properly scaled to include the range of your values. Once the points are plotted, they will generally show the trend of the graph. Finally, connect these points with a smooth, continuous curve, making sure that the curve gets closer to the horizontal asymptote as x becomes more negative, and rises swiftly as x increases. Keep in mind the end behavior; the graph should extend infinitely upwards as x goes towards positive infinity and should get indefinitely close to the x-axis as x approaches negative infinity.