Problem 49 - Algebra and Trigonometry

Question

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=\frac{1}{3} \cdot 3^{x} $$

Step-by-Step Solution

Answer

The graphs for \(f(x)=3^{x}\) and \(g(x)=\frac{1}{3} \cdot 3^{x}\) are exponential growth curves. Both share the same horizontal asymptote at y=0. Function \(f(x)=3^{x}\) is three times steeper compared to \(g(x)=\frac{1}{3} \cdot 3^{x}\).

Step 1: Graph f(x)=3^x

To graph \(f(x)=3^{x}\), plot some points using various values of x. For example, when x = -1, f(x) = 3^(-1) = 1/3. When x=0, f(x)=3^0=1, and when x=1, f(x)=3^1=3. As for the asymptote, note that as x tends towards negative infinity, \(f(x) = 3^{x}\) approaches 0. So, y=0 is a horizontal asymptote.

Step 2: Graph g(x)=1/3 * 3^x

In a similar manner, plot points using various values of x for \(g(x)=\frac{1}{3} \cdot 3^{x}\). For instance, when x=-1, g(x)=1/3 * 3^(-1) = 1/9. When x=0, g(x)=1/3 * 3^0 = 1/3, and when x=1, g(x)=1/3 * 3^1 = 1. Again, note that as x moves towards negative infinity, \(g(x)\) approaches 0. So, y=0 is a horizontal asymptote for this function as well.

Step 3: Confirm the Graphs

It can be seen that both functions get increasingly larger as x increases, but they approach 0 as x decreases. This means that they appear similar when graphed, however, \(f(x)=3^x\) is three times steeper than \(g(x)=\frac{1}{3} \cdot 3^{x}\) due to the coefficient of 1/3.

Key Concepts

Asymptotes of Functions

In the landscape of graphical analysis, the concept of asymptotes is critical to understand the behavior of functions. An asymptote can be thought of as an invisible line to which the graph of a function gets arbitrarily close but never actually touches or crosses. For exponential functions, like the ones given in our example, horizontal asymptotes are particularly significant.

Consider the function \( f(x) = 3^x \). As we explore the values of \( f(x) \) for increasingly negative values of \( x \), the result approaches zero. Mathematically speaking, we say that the function possesses a horizontal asymptote at \( y = 0 \). Similarly, the function \( g(x) = \frac{1}{3} \cdot 3^x \) also tends towards zero as \( x \) decreases, granting it the same horizontal asymptote of \( y = 0 \).

To properly identify asymptotes, examine the end behaviors of the function and observe the limits as \( x \) approaches infinity or negative infinity. In functions without horizontal asymptotes, you might encounter vertical asymptotes or slant asymptotes depending on their form. Understanding asymptotes is key to sketching the accurate long-term behavior of the function’s graph.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is the stage upon which we plot our functions. It is composed of two perpendicular lines called axes, typically labeled \( x \) and \( y \). The point where they intersect is called the origin.

When graphing functions like \( f(x) = 3^x \) and \( g(x) = \frac{1}{3} \cdot 3^x \), we place each calculated pair of \( x \) and \( y \) as a point on this system. By doing so, we translate algebraic expressions into visual representations which make it easier to grasp their behavior.

With the rectangular coordinate system, we can quickly spot where functions increase, decrease, or level off in relation to the horizontal asymptotes. This system is also handy for identifying intercepts with the axes and symmetries or periodicities, features that are pivotal to understanding the comprehensive character of a function.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software, are invaluable tools for confirming the characteristics of functions that have been graphed by hand. In our exercise example, after plotting the points and observing the horizontal asymptote for \( f(x) = 3^x \) and \( g(x) = \frac{1}{3} \cdot 3^x \), graphing utilities can be employed to confirm these findings.

These technological aids provide a quick and accurate means to visualize functions, especially when dealing with complex behaviors or when looking for precision. After inserting the function into the graphing utility, it can display the function’s graph, revealing nuances such as the precise steepness difference between the graphs of \( f(x) \) and \( g(x) \) as mentioned in the solution. Students can benefit from these utilities by contrasting their manually drawn graphs against the displayed ones, enabling them to validate their work and gain confidence in their graphing skills.