Problem 16 - College Algebra

Question

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. $$ h(x)-\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

The graph of \( h(x) = (\frac{1}{3})^{x} \) is a decreasing, smooth curve passing through the points \((-2, 9), (-1, 3), (0, 1), (1, \frac{1}{3}), (2, \frac{1}{9})\). This result is confirmed by a graphing utility.

Step 1: Creating the table of coordinates

Choose a number of values for \( x \). Good initial choices are -2, -1, 0, 1, and 2. Compute \( h(x) \) for each of these values. This calculation will give the corresponding \( y \)-values. In this case, \( h(x) = (\frac{1}{3})^{x} \), so essential coordinates will be \((-2, 9), (-1, 3), (0, 1), (1, \frac{1}{3}), (2, \frac{1}{9})\)

Step 2: Plotting the points

Plot these points on a graph. Make sure that the x-axis and y-axis are properly scaled to accommodate all the points and clearly demonstrate the pattern of the function.

Step 3: Sketching the graph

Sketch a smooth curve that passes through the plotted points. This reveals the shape of the function on the graph.

Step 4: Verifying the graph with a graphing utility

Using a graphing utility, input the function \( h(x) = (\frac{1}{3})^{x} \) and check that the graph matches with the one obtained manually. Any discrepancies can be due to inaccuracies in plotting or arithmetic. The utility can help quickly identify such errors.

Key Concepts

Exponential Functions

Understanding exponential functions is crucial when graphing specific mathematical relationships, like the one given in the exercise. An exponential function is a mathematical expression where a constant base is raised to a variable exponent, such as \[ h(x) = \left(\frac{1}{3}\right)^{x} \] In this example:

• The base is \( \frac{1}{3} \), a fractional number.
• The exponent is \( x \), which changes based on the input value, making the function dynamic.

Exponential functions have several key characteristics:

• They can model growth (when the base is greater than 1) or decay (when the base is between 0 and 1).
• The output changes multiplicatively, which means small changes in \( x \) can lead to large changes in the value of the function.
• This function decreases as \( x \) increases, showing a decay pattern due to the fractional base.

Table of Coordinates

One effective way to understand the behavior of an exponential function is by creating a table of coordinates. This involves selecting different values for \( x \) and calculating the corresponding \( h(x) \).Choosing values such as \(-2, -1, 0, 1,\) and \(2\) provides a good range of data points to observe the pattern.For example:

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

This table provides clear numerical insights:

• Positive values of \( x \) result in \( h(x) \) getting closer to zero as \( x \) increases.
• Negative and small values of \( x \) produce very large values indicating rapid growth in the opposite direction.

These calculations help in plotting the points accurately on a graph, revealing the function's structure.

Graphing Utility

A graphing utility is a powerful tool that simplifies the process of visualizing functions, including exponential ones.Using a graphing utility involves inputting the function \[h(x) = \left(\frac{1}{3}\right)^{x}\] and allowing the software to generate the corresponding graph.Some benefits of using a graphing utility include:

• Quick verification of manually drawn graphs. This helps identify any mistakes in plotting or calculations.
• The ability to explore further by adjusting axes scales or testing more values of \( x \).
• Visual clarity and precision beyond manual graphs, especially for functions involving complex behavior due to exponentially growing or decaying patterns.

Overall, these tools offer a flexible way to confirm and deepen understanding of the graph's characteristics, providing visual affirmation of theoretical insights gleaned from manual methods.