Problem 14 - 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. $$g(x)=\left(\frac{4}{3}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of function \(g(x) = (\frac{4}{3})^{x}\) is an exponential curve that increases and becomes steeper as \(x\) increases.

Step 1: Choose x-values

Select a range of numbers for \(x\), choose -2, -1, 0, 1, 2 for the calculation.

Step 2: Substitute and Calculate

Substitute chosen \(x\)-values into the function and calculate to get corresponding \(y\)-values. The table will look something like this:|-2 |\(\left(\frac{4}{3}\right)^{-2}=0.5625\)|-1 |\(\left(\frac{4}{3}\right)^{-1}=0.75\)| 0 |\(\left(\frac{4}{3}\right)^{0}=1\)| 1 |\(\left(\frac{4}{3}\right)^{1}=1.3333..\)| 2 |\(\left(\frac{4}{3}\right)^{2}=1.7777..\)

Step 3: Draw the Graph

With the table in the previous step, plot the corresponding points on the graph. Also plot the graph accurately by connecting the plotted points to form a smooth curve.

Step 4: Use Graphing Tool

Confirm your results by using a graphing tool to ensure the accuracy of the graph.

Key Concepts

Exponential Function Graph

When learning about exponential functions, it's essential to grasp how their graphs look and behave. An exponential function typically takes the form of f(x) = a^x, where a is a constant and x is the exponent. The graph of an exponential function showcases how rapidly the function's values can increase as x grows when a is greater than one. Conversely, if a is in between zero and one, the values decrease as x increases, but never reach zero.

When you draw the graph, you'll notice certain characteristics that are common to all exponential functions: the function is always positive, it increases or decreases at an ever-changing rate, and it has a horizontal asymptote usually along the x-axis. Remember, the graph will never touch or cross this asymptote. It's also important to point out that the steeper the graph, the larger the base a. Understanding these properties helps you anticipate the behavior of the graph across different ranges of x.

Coordinate Plotting

Coordinate plotting is a crucial technique used for graphing functions by hand, including exponential functions like g(x) = (4/3)^x. To plot the coordinates effectively, first, create a table of values by selecting a range of x-values, as seen in the exercise solution. Then, compute the corresponding y-values. These pairs are now your coordinates (x, y).

Using a Cartesian coordinate system, with horizontal and vertical axes typically labeled 'x' and 'y', each pair of numbers indicates a specific point on the graph. To plot these points, start with the x-value and find its location along the x-axis. From there, move vertically to the y-value that matches the given x. Once all the points are plotted on your graph, they should form a distinct pattern or shape that can then be connected to visualize the function.

Using Graphing Utilities

While hand-drawing graphs is an invaluable skill for understanding the fundamentals of functions, graphing utilities are modern tools that can assist in this process. These utilities often come as online graphing calculators or software, and they can provide a quick and accurate representation of functions, such as g(x) = (4/3)^x. To use these tools, you generally need to input the equation of the function you wish to graph.

Once the equation is entered, the graphing utility will display the curve. This allows you to confirm the correctness of your hand-drawn graph, especially if you're unsure about specific sections of it. Moreover, graphing utilities can handle much larger or difficult values that might be challenging to calculate by hand. They also showcase the behavior of the function outside the limited window of your chosen x-values, giving you a broader understanding of its characteristics.