Problem 16 - College Algebra

Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(h(x)=\left(\frac{1}{3}\right)^{x}\)

Step-by-Step Solution

Answer

The graph of the function \(h(x) = \left(\frac{1}{3}\right)^{x}\) is an exponential decay curve, decreasing from left to right. The points (-2, 9), (-1, 3), (0, 1), (1,1/3), and (2,1/9) are on the graph.

Step 1: Recognize the function

The function \(h(x)=\left(\frac{1}{3}\right)^{x}\) is an exponential function. The base of the exponential function is 1/3.

Step 2: Create a table of values

This involves choosing some values for x and calculating the corresponding values for h(x), the function. Let's choose x values such as -2, -1, 0, 1, and 2. Substituting these x-values into the function; \n For x=-2, \(h(x)= \left(\frac{1}{3}\right)^{-2}= 9\);\n For x=-1, \(h(x)= \left(\frac{1}{3}\right)^{-1}= 3\);\n For x=0, \(h(x)= \left(\frac{1}{3}\right)^{0}= 1\);\n For x=1, \(h(x)=\left(\frac{1}{3}\right)^{1}= \frac{1}{3}\);\n For x=2, \(h(x)=\left(\frac{1}{3}\right)^{2}= \frac{1}{9}\).

Step 3: Plot the points onto the graph

Take each (x, h(x)) pair from the table and plot it on a Cartesian plane. Connect the plotted points with a smooth curve. The points (-2, 9), (-1, 3), (0, 1), (1,1/3) and (2,1/9) should be on the curve.

Step 4: Confirm the graph using a graphing utility

Input the function \(h(x) = \left(\frac{1}{3}\right)^{x}\) into a graphing calculator or online graphing tool to confirm the shape and points on the graph generated in Step 3.

Key Concepts

Exponential functions

Exponential functions are a type of mathematical function where the variable is in the exponent of a constant base. In our exercise, the function is \( h(x) = \left( \frac{1}{3} \right)^x \). This base of \( \frac{1}{3} \) is especially interesting because it is a fraction less than one, which means the function will decrease, or 'decay,' as \( x \) increases. An exponential decay function exhibits a rate of decline that becomes slower over time.
Some important characteristics of exponential functions include:

• The function value is never zero, but approaches zero asymptotically as \( x \) moves towards positive infinity.
• The y-intercept in this case is always \( h(0) = 1 \) because any base raised to the zero power equals one.
• The domain of these functions is all real numbers, while the range is all positive real numbers.

Understanding these properties helps in graphing and analyzing exponential functions effectively.

Table of values

Creating a table of values is a fundamental step in graphing any function. This process involves selecting a series of \( x \) values and calculating the corresponding \( h(x) \) values for our function. For our function \( h(x) = \left( \frac{1}{3} \right)^x \), we chose the values \( x = -2, -1, 0, 1, 2 \).
For each \( x \) value, we compute \( h(x) \):

• When \( x = -2 \), \( h(x) = 9 \).
• When \( x = -1 \), \( h(x) = 3 \).
• When \( x = 0 \), \( h(x) = 1 \).
• When \( x = 1 \), \( h(x) = \frac{1}{3} \).
• When \( x = 2 \), \( h(x) = \frac{1}{9} \).

This table helps illustrate how the function behaves for different \( x \) values and provides the necessary points to plot the graph accurately. It is a preview of what the graph will look like once it is plotted on a Cartesian plane.

Graphing utility

A graphing utility is a tool, often a calculator or software, that can graph functions quickly and accurately. It is very handy for verifying your hand-drawn graphs. After plotting points manually, you can use graphing software to input \( h(x) = \left( \frac{1}{3} \right)^x \) and confirm the graph's shape and accuracy.
Using a graphing utility has several benefits:

• Immediate visual feedback of your function's behavior across different \( x \) values.
• Accuracy in plotting, which can help you catch any errors in manual calculations.
• The ability to quickly test additional \( x \) values without recalculating everything manually.

These utilities are excellent tools for learning and confirming your understanding of exponential functions and their graphs.

Cartesian plane

The Cartesian plane is a two-dimensional coordinate system where you can plot points, lines, and curves to represent functions visually. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin, \( (0, 0) \).
When graphing \( h(x) = \left( \frac{1}{3} \right)^x \), each pair \( (x, h(x)) \) from our table of values corresponds to a point on this plane:

• The point \( (-2, 9) \) shows how high the function value is when \( x \) is \(-2\).
• The point \( (0, 1) \) is the function’s y-intercept, where it crosses the y-axis.
• The points tend to get closer to the x-axis but never touch it as \( x \) increases, indicative of the asymptotic nature of exponential functions.

Connecting these points with a smooth curve shows the decay behavior as \( x \) increases, providing a visual representation of the function's behavior on the Cartesian plane.