Problem 15 - College Algebra Essentials

Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The table of values for the function \(h(x)=\left(\frac{1}{2}\right)^{x}\) are (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). When these points are plotted on a graph and a line of best fit is drawn, the result is an exponential decay curve.

Step 1: Create a Table of Values

Start by selecting a range of x-values and input them into the function to obtain corresponding y-values. For this function, let's use the x-values -2, -1, 0, 1, 2, and 3. The y-values obtained when these x-values are substituted into the function \(h(x)=\left(\frac{1}{2}\right)^{x}\) are as follows:\nh(-2) = 4,\nh(-1) = 2, \nh(0) = 1, \nh(1) = 1/2, \nh(2) = 1/4, \nh(3) = 1/8.

Step 2: Plot the Values on a Graph

On a set of axes, plot the pairs of x and y-values obtained. It's crucial to label the x-axis (horizontal axis) and y-axis (vertical axis) correctly, to avoid confusion. The points should be (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). After plotting these points, draw a curve or a line that best fits these points.

Step 3: Confirm with a Graphing Utility

Finally, use a graphing utility to graph \(h(x)=\left(\frac{1}{2}\right)^{x}\) and compare it with the hand-drawn graph. The two should match. This is a good practice to ensure that the values were calculated correctly and that the graph was correctly drawn.

Key Concepts

Coordinate Graphing

When it comes to visualizing functions, coordinate graphing is one of the most fundamental techniques in mathematics. It involves plotting points on a two-dimensional grid where each point represents a pair of values. One value is plotted along the horizontal axis, known as the x-axis, and the other value is plotted along the vertical axis, called the y-axis. To effectively graph an exponential function like

by hand, it's crucial to carefully select a range of x-values that will clearly demonstrate the behavior of the function. Once the table of values is created with corresponding y-values, you plot the points on the graph and draw a smooth curve to represent the exponential trend. Remember, attention to detail is key in this process to ensure accuracy, such as correctly labeling axes and maintaining consistent scaling.

Exponential Decay

An essential concept in graphing exponential functions is understanding exponential decay. It describes the process where quantities decrease rapidly at first and then more slowly over time, approaching zero but never actually reaching it. In the function

the base is between 0 and 1, indicating that the function represents exponential decay. The graph will show a swift drop in value for positive x-values and will approach, but not touch, the x-axis. On the other side of the y-axis, as x-values become negative, the function value increases, moving away from the x-axis. This is an important pattern to recognize when predicting the function's behavior beyond the plotted points.

Table of Values Method

The table of values method is a systematic approach to plot a function's graph precisely. After choosing a function, like you define a series of x-values that span both the positive and negative directions. Calculating the corresponding y-values using the function's formula, you create a table that pairs each x with its resultant y. This method provides a clear visual way to organize the data and get ready for plotting on the coordinate graph. It's a particularly effective strategy for manually graphing functions since it breaks down the process into manageable steps, ensuring all critical points are included to shape the graph correctly.

Graphing Utility

While manually plotting graphs is educational, technology offers tools, known as graphing utilities, to streamline this process and improve accuracy. These utilities can range from scientific calculators with graphing capabilities to sophisticated computer software. By inputting the function's formula, such as

into a graphing utility, it will automatically generate the graph. The advantages are numerous: not only can it handle complex functions that would be difficult to plot by hand, but it also allows for swift error-checking against a hand-drawn graph - as was suggested in the original exercise. As a learning strategy, using these graphing utilities reinforces your understanding of the function's shape and behavior, and prepares you for further mathematical endeavors where more complex tools become necessary.