Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. g(x) = (3/2)^x

Verified step by step guidance

1. Identify the function given: \(g(x) = \left(\frac{3}{2}\right)^x\). This is an exponential function where the base is \(\frac{3}{2}\), which is greater than 1, indicating exponential growth.
2. Create a table of values by choosing several values for \(x\), including negative, zero, and positive integers. For example, select \(x = -2, -1, 0, 1, 2\) to get a good range of points.
3. Calculate the corresponding \(g(x)\) values for each chosen \(x\) by substituting into the function: \(g(x) = \left(\frac{3}{2}\right)^x\). Remember that for negative exponents, \(a^{-n} = \frac{1}{a^n}\).
4. Plot the points \((x, g(x))\) from your table on a coordinate plane. Since the function is exponential growth, the graph should increase as \(x\) increases and approach zero but never touch the \(x\)-axis as \(x\) decreases.
5. Use a graphing utility to input \(g(x) = \left(\frac{3}{2}\right)^x\) and compare the graph with your hand-drawn points to confirm accuracy and understand the shape of the exponential curve.

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 g(x) = (3/2)^x grows as x increases because the base 3/2 is greater than 1, resulting in exponential growth.

Creating a Table of Coordinates

To graph a function by hand, select various x-values, substitute them into the function, and calculate the corresponding y-values. Plotting these (x, y) pairs on a coordinate plane helps visualize the function's shape and behavior.

Using Graphing Utilities

Graphing utilities, such as calculators or software, allow quick and accurate plotting of functions. They help confirm the accuracy of hand-drawn graphs and provide insights into features like intercepts, asymptotes, and growth trends.