Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. h(x) = (1/2)^x

Verified step by step guidance

1. Identify the function given: \(h(x) = \left(\frac{1}{2}\right)^x\). This is an exponential function with base \(\frac{1}{2}\), which means it is a decay function because the base is between 0 and 1.
2. Create a table of values by choosing several values for \(x\), including negative, zero, and positive integers. For example, select \(x = -2, -1, 0, 1, 2\).
3. Calculate the corresponding \(h(x)\) values for each chosen \(x\) by substituting into the function: \(h(x) = \left(\frac{1}{2}\right)^x\). Remember that negative exponents mean taking the reciprocal, so \(\left(\frac{1}{2}\right)^{-x} = 2^x\).
4. Plot the points \((x, h(x))\) from your table on a coordinate plane. This will help you visualize the shape of the graph.
5. Draw a smooth curve through the plotted points, noting that the graph approaches the \(x\)-axis but never touches it (the \(x\)-axis is a horizontal asymptote). Optionally, use a graphing utility to confirm the accuracy of your hand-drawn graph.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. In this question, h(x) = (1/2)^x is an exponential decay function because the base is between 0 and 1. Understanding how the function behaves as x increases or decreases is essential for graphing.

Creating a Table of Coordinates

To graph a function by hand, select various x-values and compute the corresponding y-values to form coordinate pairs. This table helps visualize key points on the graph, showing how the function changes and providing a foundation for sketching the curve accurately.

Using Graphing Utilities

Graphing utilities, such as calculators or software, allow you to plot functions quickly and verify hand-drawn graphs. They help confirm the shape, intercepts, and behavior of the function, ensuring accuracy and reinforcing understanding of the function's properties.