Problem 52 - 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. $$g(x)=e^{0.5 x}-1$$

Step-by-Step Solution

Answer

The function \(g(x)=e^{0.5 x}-1\) can be sketched by drawing a curve through a set of points determined by a table of values. The graph has a horizontal asymptote at y=-1, the graph tends towards this line as x approaches negative infinity.

Step 1: Construct a Table of Values

To create a table of values for the function, a useful approach is to choose a set of x-values and then calculate the corresponding y-values by substituting each x-value into the function equation. These calculations can be made using a calculator or other graphing utility. For instance, one could choose x-values of -2, -1, 0, 1, and 2, and substitute each of these into the function \(g(x)=e^{0.5 x}-1\) to get the corresponding y-values.

Step 2: Sketch the Graph

With the table of values, we can now proceed to plot these points and sketch a graph. The x-values chosen are along the horizontal axis and the y-values are on the vertical axis. After plotting the points, draw a smooth curve through the points to represent the function on the graph.

Step 3: Identify Asymptotes

Asymptotes are lines to which a function approaches as we move towards infinity or negative infinity. In the case of the given function \(g(x)=e^{0.5 x}-1\), the horizontal asymptote can be identified by looking at the behavior of the function as x approaches negative infinity. If there was a vertical asymptote, it would be identified by looking where x makes the function undefined, which doesn't happen in the case of this exponential function

Key Concepts

Table of Values

Understanding the table of values is crucial for graphing functions accurately. To start, select a range of x-values that you're interested in, typically including both negative and positive values to observe different behaviors of the function. For the exponential function g(x) = e^{0.5x} - 1, you might choose x-values like -2, -1, 0, 1, and 2. Using a calculator, compute the corresponding y-values by substituting x into the function. For example, when x=0, g(0) = e^{0.5(0)} - 1 = 0.

Repeat this process for each chosen x-value and tabulate your results. This tabulation facilitates a visual interpretation when you plot these coordinates on a graph. The aim is for students to recognize patterns and behavior of an exponential function's output as its input changes, making the relationship between variables more comprehensible.

Sketching Graphs

Once you have a table of values, you're ready to sketch the graph. Begin by plotting each (x, y) pair from your table on a coordinate plane. When graphing g(x) = e^{0.5x} - 1, you'll notice that as x increases, the y-values grow rapidly, reflecting the exponential increase. Conversely, as x decreases, the y-values approach -1, but never reach it.

After plotting the points, draw a smooth curve that best fits them. For this exponential function, the curve will start close to the horizontal axis on the left and rise steeply to the right. This visual representation helps students intuitively grasp the function's growth rate, allowing for better prediction and understanding of exponential behavior beyond the plotted points.

Identifying Asymptotes

Horizontal Asymptotes

For our function g(x) = e^{0.5x} - 1, the horizontal asymptote represents a y-value that the function approaches but never quite reaches as x heads towards negative infinity. In this case, as x becomes very large in the negative direction, e^{0.5x} approaches 0 and thus g(x) gets closer and closer to -1. This means our horizontal asymptote is the line y = -1.

No Vertical Asymptotes

On the other hand, vertical asymptotes occur where the function heads towards infinity or negative infinity, typically at points of discontinuity. For this exponential function, there are no vertical asymptotes because it's defined for all real numbers of x and does not encounter any discontinuities or 'infinities'.

Exponential Function Properties

Exponential functions like g(x) = e^{0.5x} - 1 possess specific properties that define their behavior. One key property is that they are always growing or decaying, never staying constant for different values of x. Specifically, g(x) shows exponential growth because its base, e, is greater than 1. Another property is that the horizontal asymptote acts as an invisible boundary the function will never cross.

Furthermore, exponential functions are continuous with no breaks, jumps, or holes. Their domains are all real numbers, meaning they have an input (x) for every point along the x-axis. For the range, these functions will yield all real y-values above or below the horizontal asymptote, depending on whether it's growth or decay. Understanding these properties enhances students’ ability to anticipate the function's behavior, even for x-values not included in their table of values.