Problem 15 - 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. $$ h(x)-\left(\frac{1}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = \left(\frac{1}{2}\right)^{x}\) is a decreasing curve starting at the y-axis with a value of 1. This is confirmed by both the hand-drawn graph and the graph created with a graphing utility.

Step 1: Understand the function

The function given is \(h(x) = \left(\frac{1}{2}\right)^{x}\). This is an exponential function, where the base is \(\frac{1}{2}\). This function will decrease as the x-value increases.

Step 2: Create a table of coordinates

Choose some x-values and plug them into the function to produce corresponding y-values. Here, the table could be for example:| x | -2 | -1 | 0 | 1 | 2 ||---|----|----|---|---|---|| h(x) | 4 | 2 | 1 | 0.5 | 0.25 |The coordinates obtained from this table will be used to hand draw the graph.

Step 3: Graph the function

Plot the coordinates obtained from the table on a graph, using x-values for the horizontal axis, and h(x)-values for the vertical axis. Connect the dots to form a smooth curve. Make sure to label each axis.

Step 4: Confirm the graph with a graphing utility

Use a graphing utility to plot the function \(h(x) = \left(\frac{1}{2}\right)^{x}\). Compare this graph with the hand-drawn graph to ensure accuracy.

Key Concepts

Graphing Utility

A graphing utility is a tool that allows us to visualize mathematical functions easily. It can be a physical tool like a graphing calculator or a software application like Desmos or GeoGebra. These tools are designed to help both students and educators verify their hand-drawn graphs for accuracy.

Here’s how a graphing utility works:

• Input the function into the utility.
• It automatically generates a graph.
• You can adjust the viewing window to zoom in or out, helping you see more details.

Using a graphing utility is especially helpful when working with complex functions or ensuring that your hand-drawn graph accurately represents the function. It provides a way to check that every calculated point lies on the graph correctly.

Coordinate Table

Creating a coordinate table is a simple yet effective way to visualize a function's behavior. For exponential functions, like our function \(h(x) = \left(\frac{1}{2}\right)^{x}\), this means selecting a range of x-values and calculating the corresponding y-values.

Here's how you can create a coordinate table:

• Select a range of x-values, both negative and positive, to observe changes in the function.
• Calculate their y-values using the function formula. For instance, inputs like -2, -1, 0, 1, and 2 can help reveal how the function behaves.

A coordinate table not only helps in plotting the graph but also in understanding the rate of growth or decay of the function. It offers a structured way to see how each input affects the output.

Function Graphing

Graphing a function is turning numerical data into a visual representation. For instance, in the exponential function \(h(x) = \left(\frac{1}{2}\right)^{x}\), each coordinate pair from the table is plotted on the coordinate plane representing the x and h(x) values.

To graph a function:

• Start by drawing x and y axes and labeling them.
• Plot each coordinate point on the graph as a dot.
• Connect the points with a smooth curve, making note of the curve’s direction and how it visually represents the function's behavior.

Through graphing, we can see the nature of exponential decay firsthand, understanding its progression and trend lines.

Exponential Decay

Exponential decay refers to the process by which a quantity decreases at a rate proportional to its current value. The function \(h(x) = \left(\frac{1}{2}\right)^{x}\) is a classic example of exponential decay, given that the base of the exponential term, \(\frac{1}{2}\), is between 0 and 1.

This means that as x increases:

• The function's value decreases (decay) because the exponentiation of a fraction leads to smaller and smaller values.
• The graph will show a downward trend as the x-values move from left to right.

Understanding exponential decay is fundamental in various fields, such as physics for radioactive decay or finance for depreciating assets. This concept helps us describe real-world situations where quantities diminish over time.