Problem 12 - 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. $$f(x)=5^{x}$$

Step-by-Step Solution

Answer

Plot the points derived from the table of values and draw an increasing curve which represents the function \(f(x)=5^x\). This curve has a horizontal asymptote at \(y=0\) (x-axis) and passes through \(y=1\) when \(x=0\). A graphing utility can be used to check the accuracy of the resulting plot.

Step 1: Understand the Function

The function is \(f(x)=5^{x}\), an exponential function where the base is greater than 1, which means that its graph should show exponential growth. If the base was between 0 and 1, the function would show exponential decay.

Step 2: Create a Table of Values

Next, we choose several values for \(x\) (both negative and positive) and calculate the corresponding \(f(x)\) values. Use this table to plot points on the graph.

Step 3: Plot Points and Draw the Graph

Using the values from the table, plot points on a graph. As \(x\) increases, \(f(x)\) will also increase and as \(x\) decreases, \(f(x)\) will asymptotically approach 0, but never reach it.

Step 4: Confirm Graph Using a Graphing Utility

Using a graphing calculator or an online tool plug in \(f(x)=5^{x}\) to verify the accuracy of the hand-drawn graph.

Key Concepts

Graphing Functions

Graphing a function is the process of visually representing its behavior on a coordinate plane. For functions like \( f(x)=5^{x} \), which is an example of an exponential function, understanding the shape and direction of the graph can help you see how the function operates. Exponential functions such as \( 5^x \) tend to rise quickly as \( x \) becomes larger, reflecting exponential growth. The critical point to note while graphing such functions is that they show rapid increases, and the graph should pass through the point \( (0,1) \), since any nonzero number raised to the power 0 equals 1.
To effectively graph a function, sketch a curve that reflects its increasing nature, verifying that it doesn't incorrectly loop down or flatten without demonstrating growth.

Coordinate Tables

A coordinate table is a useful tool when graphing any function. To create a coordinate table for the function \( f(x) = 5^x \), select several \( x \) values - ideally a mix of negative, zero, and positive values - and compute the corresponding \( f(x) \). For instance:

• For \( x = -2 \), \( f(x) = 5^{-2} = \frac{1}{25} \)
• For \( x = 0 \), \( f(x) = 5^{0} = 1 \)
• For \( x = 1 \), \( f(x) = 5^{1} = 5 \)
• For \( x = 2 \), \( f(x) = 5^{2} = 25 \)

Each pair \((x, f(x))\) becomes a point on your graph. By plotting these points accurately, you will witness the quick ascent of the exponential curve.
This process not only reinforces how exponential growth behaves but also makes sure your hand-drawn graph is accurate.

Exponential Growth

Exponential growth refers to the pattern of data that increases rapidly as time goes on. In the function \( f(x)=5^x \), this characteristic is shown clearly; each increase in \( x \) results in a proportionate increase in \( f(x) \). It's a rapid form of growth that can be observed in various contexts, such as populations, investments, and technologies. The graph of an exponential growth function begins with a slower incline, which becomes steeper with higher \( x \) values.
Recognizing exponential growth within the graph helps understand the power of compounding, which is a fundamental principle in growth systems. As \( x \) increases, the rate of \( f(x) \) also rises significantly faster, which distinguishes exponential functions from linear or polynomial functions.

Using Graphing Utilities

Graphing utilities are digital tools that provide an accurate picture of your function's behavior quickly. They are particularly useful for validating hand-drawn graphs. By entering the function \( f(x) = 5^x \) into a graphing calculator or software, you can instantly see the true graph of the function. This can help you spot errors in your hand-drawn attempts.
These tools also provide advantages like zoom features, which allow for examination of specific graph parts, and axis adjustment, to help visualize subtle changes in growth at different scales. Using graphing utilities can save time, and make complex graphs more accessible.