In Exercises 11–18, graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. f(x) = 4^x

Verified step by step guidance

1. Identify the function given: \(f(x) = 4^{x}\). This is an exponential function where the base is 4 and the exponent is the variable \(x\).
2. Create a table of values by choosing several values for \(x\), including negative, zero, and positive values. For example, select \(x = -2, -1, 0, 1, 2\).
3. Calculate the corresponding \(f(x)\) values for each chosen \(x\) by evaluating \$4^{x}\(. For instance, when \)x = 0\(, \)f(0) = 4^{0}$.
4. Plot the points \((x, f(x))\) from your table on a coordinate plane. This will help visualize the shape of the graph.
5. Use a graphing utility to input \(f(x) = 4^{x}\) and compare the graph it produces with your hand-drawn points to confirm accuracy.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. The function grows or decays rapidly depending on whether a is greater than or less than 1. Understanding this helps in predicting the shape and behavior of the graph.

Creating a Table of Coordinates

To graph a function by hand, select various x-values and compute their corresponding f(x) values. Plotting these (x, f(x)) points on the coordinate plane provides a visual representation of the function’s behavior, which is essential for sketching an accurate graph.

Using Graphing Utilities

Graphing utilities, such as calculators or software, allow you to quickly plot functions and verify hand-drawn graphs. They help confirm accuracy and provide insight into the function’s features like intercepts, growth rate, and asymptotes.