Problem 48 - College Algebra Essentials

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

The graphs of functions \(f(x) = 3^x\) and \(g(x) = -3^x\) are reflections of each other over the x-axis, with the x-axis being a horizontal asymptote for both, so the equation of the asymptote is \(y = 0\).

Step 1: Understand the functions

The functions that need to be graphed are exponential functions \(f(x) = 3^x\) and \(g(x) = -3^x\). Exponential functions have a shape that rapidly increases or decreases depending on its base, which is 3 in this case.

Step 2: Identify the nature and behavior

For the function \(f(x) = 3^x\), as x approaches negative infinity (\(-\infty\)), the function approaches zero but never touches zero. Therefore, the x-axis is a horizontal asymptote for \(f(x)\). By the same reasoning, for the function \(g(x) = -3^x\), as x approaches positive infinity (\(+\infty\)) the function also approaches zero, so the x-axis is also a horizontal asymptote for \(g(x)\).

Step 3: Graph the functions

Start by creating a table of values for both \(f(x)\) and \(g(x)\), or use a graphing calculator if allowed. Draw two curves with x-axis as their asymptotes. The curve of \(f(x)=3^x\) is in the first and second quadrants, while the curve of \(g(x)=-3^x\) is in the third and fourth quadrants. Additionally, these two functions are reflections of each other over the x-axis.

Step 4: Write the equations of the asymptotes

Since horizontal asymptotes exist for both functions and are along the x-axis, the equation for the asymptotes is \(y = 0\).

Key Concepts

Graphing Functions

Graphing a function is all about visualizing the equation on a plane so that we can easily understand its behavior. In this exercise, the functions given are exponential, specifically \(f(x) = 3^x\) and \(g(x) = -3^x\), which stand out due to their rapid rate of increase or decrease.

• Exponential functions have characteristic curves. For \(f(x) = 3^x\), the graph will shoot upwards as \(x\) becomes larger.
• The graph of \(g(x) = -3^x\) is similar, but flipped over the x-axis because of the negative sign.

Considering this, graphing these functions involves plotting the corresponding points and drawing smooth curves through them. Using a graphing calculator can help in accurately sketching the steepness and direction of the curves. Just don't forget to include the behavior as they move towards infinity, which relates to asymptotes.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches, but never actually reaches as the inputs or \(x\) values become very large (positive or negative). This concept is essential when understanding exponential functions. For \(f(x) = 3^x\), as \(x\) approaches negative infinity, \(f(x)\) gets closer and closer to 0, but never quite reaches it. This means the x-axis, or \(y = 0\), is a horizontal asymptote. Similarly, for \(g(x) = -3^x\), as \(x\) approaches positive infinity, the function also moves nearer to zero. Thus, for both \(f(x)\) and \(g(x)\), the equation of the horizontal asymptote is \(y = 0\).

• Remember, horizontal asymptotes describe end behavior for large \(x\) values.
• They highlight that while the function values get extremely tiny, they never flat out become zero.

Rectangular Coordinate System

The Rectangular Coordinate System, also known as the Cartesian plane, is pivotal for graphing any function. This system is composed of an \(x\)-axis (horizontal) and a \(y\)-axis (vertical) that intersect at the origin (0, 0). Here's how it aids our graphing:

• The x-axis helps determine left or right movements while plotting points on the graph.
• The y-axis assesses upward or downward movements.
• Each point on the graph corresponds to an \((x, y)\) pair.

For this exercise, \(f(x) = 3^x\) and \(g(x) = -3^x\) cover both the positive and negative sides of these axes. \(f(x)\) appears in the first and second quadrants, whereas \(g(x)\) occupies the third and fourth quadrants. Understanding how these functions occupy different quadrants helps in anticipating their behavior, especially when sketching by hand.