Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The function \( h(x) = (\frac{1}{2})^{x} \) can be graphed using a table of coordinates and then plotted on a graph. The coordinates at x values of -2, -1, 0, 1, and 2 are (-2, 4), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25)

Step 1: Identify the Function

Identify the function that needs to be graphed. In this case, the function is \( h(x) = (\frac{1}{2})^{x} \).

Step 2: Create a Table of Coordinates

A table of coordinates is created by choosing a range of values for 'x' and then calculating the corresponding 'h(x)' values. For example, let's choose x values as -2, -1, 0, 1, and 2. Then, we find their corresponding h(x) values using the function \( h(x) = (\frac{1}{2})^{x} \). This gives a table with values (-2, 4), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25).

Step 3: Plot the Points and Draw the Graph

After obtaining the table of coordinates, plot these points on a coordinate graph. After plotting these points, draw a curve passing through these points to represent the function \( h(x) = (\frac{1}{2})^{x} \).

Step 4: Optionally, Verify with a Graphing Utility

After completing the graph, it's an option to use a graphing utility to check the work. Enter the function \( h(x) = (\frac{1}{2})^{x} \) into the graphing utility and compare the generated graph to the hand-drawn graph. They should look the same if all steps have been conducted correctly.

Key Concepts

Coordinate Graphing

Graphing functions is a fundamental part of algebra and helps in visualizing the behavior of mathematical relationships. The process involves plotting points on a coordinate plane which is a two-dimensional surface with a horizontal axis (x-axis) and a vertical axis (y-axis). To graph an exponential function such as \( h(x) = (\frac{1}{2})^{x} \), you start by making a table of coordinates.

For instance, you select values for \( x \), substitute these into the function to get \( h(x) \) values, and then plot these ordered pairs on the graph. These points represent specific locations on the plane, where the 'x' value determines the position left or right of the vertical axis, and the 'h(x)' value determines the position above or below the horizontal axis. The ordered pairs from the exercise like (0, 1), (1, 0.5), and (2, 0.25), once plotted, will form a distinct pattern showing the nature of exponential decay.

Exponential Decay

Exponential decay refers to the process of decreasing rapidly at a rate proportional to the value's current size. In the context of the function \( h(x) = (\frac{1}{2})^{x} \), the base of the exponential, which is \( \frac{1}{2} \), is less than 1. This tells us that as \( x \) increases, \( h(x) \) will decrease, approaching zero but never actually reaching it. This is due to the nature of exponential decay where the function's value is halved every time \( x \) is increased by 1.

Understanding exponential decay is key in various scientific fields such as radioactive decay in physics, depreciation in finance, and population decline in biology. In graphing, the curve will fall steeply and then level off, approaching the x-axis as a horizontal asymptote. The graph will never touch the x-axis, illustrating the perpetual but diminishing values.

Graphing Utility Verification

Once a function is graphed by hand, it is often helpful to verify your work with a graphing utility, which is a tool that can graph mathematical functions with great precision. This step is not compulsory but serves as a good check to ensure the manual graph aligns with the actual function. For the given function \( h(x) = (\frac{1}{2})^{x} \), entering it into a graphing utility will produce a graph rapidly declining towards the x-axis.

A correct hand-drawn graph should closely match the one generated by the graphing utility. This comparison helps in catching any potential errors made during plotting or while connecting the points. It can also be a learning opportunity to better understand how different functions behave and how small changes in the function's equation may produce significant changes in its graph.