Problem 48 - 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. $$ f(x)=3^{x} \text { and } g(x)=-3^{x} $$

Step-by-Step Solution

Answer

Both functions \( f(x) = 3^{x} \) and \( g(x) = -3^{x} \) have the line\( y = 0 \) as their horizontal asymptote. The graph of \( f(x) = 3^{x} \) starts from the point (0,1) and rises, whereas the graph of \( g(x) = -3^{x} \) starts at (0,-1) and drops.

Step 1: Graphing f(x) = 3^{x}

The graph of \( f(x) = 3^{x} \) is a curve that starts from the point (0,1) and rises on the right. It gets infinitely close to the x-axis as x decreases, but does not touch or cross it.

Step 2: Identifying Asymptote of f(x) = 3^{x}

The x-axis, namely the line \( y = 0 \), serves as a horizontal asymptote for the function \( f(x) = 3^{x} \). This means that as x goes to negative infinity, f(x) will get closer and closer to zero but will never reach it.

Step 3: Graphing g(x) = -3^{x}

The function \( g(x) = -3^{x} \) is a reflection of \( f(x) = 3^{x} \) across the x-axis. Hence, it also starts at the point (0,-1) and drops down to the right. Likewise, it gets infinitely close to the x-axis as x decreases without touching or crossing it.

Step 4: Identifying Asymptote of g(x) = -3^{x}

Similar to the previous step, the function \( g(x) = -3^{x} \) has the x-axis (line \( y = 0 \)) as its horizontal asymptote. As x goes to negative infinity, g(x) approaches, but never reaches zero.

Step 5: Confirming Graphs Using a Graphing Utility

To confirm these hand-drawn graphs, you could use any graphing utility such as an online graphing calculator or software like Desmos or GeoGebra. Inputting the equations will display their respective curves and asymptotes.

Key Concepts

Horizontal Asymptote

When graphing exponential functions like f(x) = 3^{x} and g(x) = -3^{x}, you will encounter an important feature known as a horizontal asymptote. This is a horizontal line that the graph approaches but never quite touches, regardless of how far the graph stretches along the x-axis. In these cases, the horizontal asymptote is the x-axis itself, or mathematically, y = 0. The function's value gets infinitely close to zero as the variable x heads towards negative infinity, setting a boundary that the graph will never cross. This concept shines a light on the behavior of exponential functions at their extremes and guides us in sketching accurate representations.

Understanding horizontal asymptotes is crucial when calculating limits at infinity and also has practical implications in fields like economics and science, where they can represent thresholds that are approached but never fully reached.

Rectangular Coordinate System

The rectangular coordinate system is a foundational tool for graphing functions like f(x) = 3^{x} and g(x) = -3^{x}. It consists of two perpendicular lines, usually labeled as the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin (0,0). Exponential functions are plotted within this coordinate system by marking a series of points whose coordinates correspond to value pairs (x, f(x)) and then connecting these points to form the graph.

For instance, f(x) = 3^{x} starts at (0,1) because f(0) = 3^{0} = 1, and moves upwards and to the right. This system not only aids in visualizing the properties of functions but also in solving geometrical problems and interpreting graphical data, making it an indispensable tool in mathematics and other scientific disciplines.

Exponential Growth and Decay

The functions f(x) = 3^{x} and g(x) = -3^{x} demonstrate two key phenomena in mathematics: exponential growth and decay. When the base of an exponent is greater than 1, as in f(x) = 3^{x}, the function models exponential growth as x increases. This rapid escalation is a telltale of exponential growth, as seen in fields like finance, biology, and physics, where variables can multiply at a constant rate over time.

Reflection Across the x-axis

The scenario shifts when considering g(x) = -3^{x}. It is the reflection of f(x) across the x-axis, resulting in exponential decay for positive values of x. The 'negative' in front of the base causes the function to dip below the x-axis, creating a mirror image of the growth function above the x-axis. This concept helps in understanding phenomena such as radioactive decay or cooling processes, where a quantity diminishes over time at a rate proportional to its current value.

Reflection Across the x-axis

Reflection across the x-axis is a transformation that alters the graph of a function by flipping it over the x-axis. For example, g(x) = -3^{x} is the reflection of f(x) = 3^{x} across the x-axis. Each point (x, y) on the graph of f(x) is mapped to (x, -y) on g(x). As such, while the original function f(x) is rising upwards from the point (0,1), the reflection g(x) falls downwards starting at (0,-1).

This mirror-effect illustrates how powerful simple transformations can be in understanding and creating graphs. When you understand reflection, you can predict the behavior of related functions and solidify your grasp of the symmetrical properties inherent in mathematical representations.