Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Select integers from \(-2\) to 2 , inclusive, for \(x\). Then describe how the graph of g is related to the graph of \(f .\) 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 graph of \(g(x)\) is a reflection of the graph of \(f(x)\) across the line \(y=1\)

Step 1: Define the Function \(f(x)\)

First, define the function \(f(x)=3^{x}\) and create a table of values for \(f(x)\) when \(x\) is from -2 to 2 in increments of 1. The table of values for \(f(x)\) is: \((-2, 3^{-2}=0.111) ,(-1, 3^{-1}=0.333) ,(0, 3^{0}=1) ,(1, 3^{1}=3) ,(2, 3^{2}=9)\)

Step 2: Define The Function \(g(x)\)

Next, define the function \(g(x)=3^{-x}\) and create a table of values for \(g(x)\) using the same set of \(x\) values. The table of values for \(g(x)\) is: \((-2, 3^{2}=9) ,(-1, 3^{1}=3) ,(0, 3^{0}=1) ,(1, 3^{-1}=0.333) ,(2, 3^{-2}=0.111)\)

Step 3: Graph The Functions \(f(x)\) and \(g(x)\)

Plot the tables of values for both \(f(x)\) and \(g(x)\) on the same graph. The graph of \(f(x)\) will be increasing as \(x\) increases while the graph of \(g(x)\) will be decreasing as \(x\) increases. The two functions are reflections of each other across the line \(y=1\).

Step 4: Describe The Relationship Between The Graphs of \(f(x)\) and \(g(x)\)

After graphing the two functions, compare them. The graph of \(g(x)\) is a reflection of \(f(x)\) in the line \(y=1\). If \(f(x)\) goes up, \(g(x)\) goes down and vice versa.

Step 5: Confirm With A Graphing Utility

Use a graphing utility to confirm the hand drawn graphs. The shapes of the graphs should confirm to our expectations, with \(f(x)\) increasing and \(g(x)\) decreasing.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable is in the exponent. For instance, in the function \(f(x) = 3^x\), the base is constant (3) and the exponent is the variable \(x\). This specific function represents exponential growth, as the value of \(f(x)\) increases rapidly with increasing values of \(x\).

When graphing exponential functions, the shape of the graph typically starts off gently, near the horizontal axis, and then rises or falls steeply, depending on whether the base is greater than one (growth) or between zero and one (decay). For the function \(f(x) = 3^x\) specifically, the graph will always pass through the point \((0,1)\), because any nonzero number raised to the power of zero equals one. It's also important to note that the graph of an exponential function never touches the x-axis, approaching it but never reaching zero.

Coordinate System

A coordinate system in math is a way to determine each point uniquely in a plane by a pair of numerical coordinates. The most commonly used system is the rectangular coordinate system, also known as the Cartesian coordinate system. It's made up of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).

The intersection where the x-axis and y-axis meet is called the origin, which has coordinates (0,0). Each point on the plane can be represented by an ordered pair of numbers (x, y), which indicate the positions along the x-axis and y-axis respectively. When graphing functions such as \(f(x) = 3^x\) and \(g(x) = 3^{-x}\), we use the coordinate system to plot points that we derive from the functions' values and then connect these points to form the graph.

Function Transformation

Function transformations involve altering the appearance of the original graph of a function in various ways, including shifting, stretching, compressing, and reflecting. These transformations allow us to understand how the graph of one function relates to another.

For example, suppose you have a base function \(f(x)\) and you're considering the function \(g(x) = 3^{-x}\). Since \(g(x)\) can be derived by replacing \(x\) with \(-x\) in \(f(x)\), this represents a reflection across the y-axis. This is a horizontal reflection. Moreover, despite these transformations, the new graph will retain certain characteristics of the original function's graph, such as the intercepts with the axes or the asymptotes, based on the type of transformation applied.

Reflection Across a Line

Reflection across a line is a type of transformation that 'flips' the graph over a specified line. Imagine you were to place a mirror along that line—the original graph and its reflection would be symmetric with respect to the mirror.

In the case of the functions \(f(x) = 3^x\) and \(g(x) = 3^{-x}\), the reflection occurs across the line \(y=1\). This means for any value of \(x\), the y-value of \(g(x)\) is the inverse of the y-value of \(f(x)\). Hence, where the graph of \(f(x)\) rises, the graph of \(g(x)\) will fall in a symmetrical fashion with respect to the line \(y=1\). Reflecting a function's graph in this way is a powerful visual tool to explore how two functions relate to each other across a line of symmetry.