Problem 27 - 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. $$f(x)=2^{x} \text { and } g(x)=2^{x-2}$$

Step-by-Step Solution

Answer

Observing the plots of both functions \(f(x)\) and \(g(x)\), it is evident that the graph of \(g\) is a horizontal translation of the graph of \(f\), shifted 2 units to the right.

Step 1: Understand the functions

Firstly, observe the given functions \(f(x) = 2^x\) and \(g(x) = 2^{x-2}\). Both functions are exponential functions, but the difference is that function \(g\) is shifted two units to the right compared to function \(f\). This shift results from the \(-2\) that is subtracted from \(x\) in \(g(x) = 2^{x-2}\).

Step 2: Determine the points for the plots

Now, choose integers from \(-2\) to \(2\) for \(x\) as instructed, and distinguish these points for both functions. For \(f(x) = 2^x\), with \(x\) running from \(-2\) to \(2\), we get points \((-2, 1/4), (-1, 1/2), (0,1), (1,2), (2,4)\). For \(g(x) = 2^{x-2}\), we get points \((-2, 1/16), (-1, 1/8), (0,1/4), (1,1/2), (2,4)\).

Step 3: Plot the graphs

Having obtained the points, the next step is to plot these points on the same graph, which will allow us to visualize the graphs of both functions.

Step 4: Describe the relationship

By examining the plots, it is clear that the graph of function \(g(x) = 2^{x-2}\) is similar to the graph of \(f(x) = 2^x\), with the only difference being that \(g\) is shifted 2 units to the right. This shift corresponds to the \(-2\) that is subtracted from \(x\) in \(g(x)\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. In these functions, the base (which is often Euler's number, but in this case is 2) is raised to the power of a variable. For example, in the function \( f(x) = 2^x \), 2 is the base and \( x \) is the exponent, which varies the output of the function.

These functions are known for their rapid growth as the variable increases. They have applications in many fields such as finance for compound interest calculations, in biology for population models, and in physics for radioactive decay.

A unique property of exponential functions is their consistent growth rate, which multiplies by the same factor over equal increments. This characteristic makes them extremely important for modeling real-life scenarios that involve exponential growth or decay.

Plotting Points

Plotting points is an essential step in graphing any function. It involves determining specific coordinates from the function's equation and marking these on a graph.

For the function \( f(x) = 2^x \), and considering a plot ranging from \( x = -2 \) to \( x = 2 \), you calculate the value of \( f(x) \) for each integer within this range. The corresponding outputs are paired with their respective inputs to form coordinates or points (e.g., \((-2, \, 0.25), (1, \, 2)\)).

This same process is repeated for the function \( g(x) = 2^{x-2} \), resulting in its own set of points. By plotting these points on a coordinate system, you begin to trace out the path of the function, gaining insight into its behavior over the specified domain.

Graphing Utility

A graphing utility is a tool, often a calculator or a software program, that assists in plotting mathematical graphs and visualizing complex functions.

For exponential functions like \( f(x) = 2^x \) and \( g(x) = 2^{x-2} \), using a graphing utility can be incredibly helpful. It not only provides a visual confirmation of your hand-drawn graphs but also helps detect any errors.

Graphing utilities can save time and improve accuracy, especially when dealing with intricate or large-scale functions. They often include functionalities like zooming in, adjusting axes scales, and plotting multiple functions simultaneously, thus offering deeper insights and a more comprehensive understanding of the graphical relationship between functions.

Coordinate System

The coordinate system is a network of horizontal and vertical lines used to determine the position of points. When graphing functions like \( f(x) = 2^x \) and \( g(x) = 2^{x-2} \), a rectangular coordinate system (also known as the Cartesian coordinate system) is most commonly used.

This system is characterized by its x-axis (horizontal) and y-axis (vertical), which intersect at the origin (0,0). Each point on a graph is represented by an ordered pair \((x, y)\) indicating its horizontal and vertical position relative to the origin.

Understanding how to navigate and employ the coordinate system is crucial for accurately plotting functions. It provides a structured framework within which mathematical relationships can be visually interpreted and analyzed, allowing for clear visual comparisons between different functions like assessing how much \( g(x) = 2^{x-2} \) shifts relative to \( f(x) = 2^x \).