Problem 32 - 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. $$f(x)=2^{x} \text { and } g(x)=2^{x}-1$$

Step-by-Step Solution

Answer

The graph of \(g(x) = 2^{x} - 1\) is a vertical shift down by 1 unit from the graph of \(f(x) = 2^{x}\) in the given range of x-values.

Step 1: Determine the y-values for \(f(x) = 2^x\)

Given the range of \(x\) values as -2, -1, 0, 1, and 2, we can calculate the corresponding \(y\) values by plugging \(x\) into the function \(f(x)\). Hence, for \(f(x) = 2^x\), the corresponding ordered pairs will be:\n(-2, \(2^{-2}\)), (-1, \(2^{-1}\)), (0, \(2^0\)), (1, \(2^1\)), (2, \(2^2\)). These evaluate to: (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), and (2, 4).

Step 2: Determine the y-values for \(g(x) = 2^x - 1\)

Similarly, calculate the \(y\) values for function \(g(x)\) by substituting each \(x\) value into the equation \(g(x) = 2^x - 1\). The resulting ordered pairs for \(x\) values -2, -1, 0, 1, and 2 will be:\n(-2, \(2^{-2} - 1\)), (-1, \(2^{-1} - 1\)), (0, \(2^0 - 1\)), (1, \(2^1 - 1\)), and (2, \(2^2 - 1\)). After calculation, these become: (-2, -0.75), (-1, -0.5), (0, 0), (1, 1), and (2, 3).

Step 3: Graph the functions

Using the rectangular coordinate system, plot the points calculated in the previous steps for both functions \(f(x)\) and \(g(x)\). The functions \(f\) and \(g\) will be distinct lines on this coordinate system.

Step 4: Analyse the graphs

After plotting the graphs for the functions \(f(x) = 2^{x}\) and \(g(x) = 2^{x} - 1\), you can observe that the graph of \(g(x)\) is a vertical shift down by 1 unit from the graph of \(f(x)\).

Step 5: Confirm using graphing utility

As instructed, use a graphing calculator or other appropriate software to plot these functions on the same graph and confirm your findings. The electronic graph should validate that function \(g\) is indeed a vertical shift of function \(f\) down by one unit.

Key Concepts

Rectangular Coordinate System

Understanding the rectangular coordinate system is essential for graphing exponential functions. It's comprised of two perpendicular lines; the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. These axes intersect at a point called the origin, denoted as (0,0).

Each point in this system is represented by an ordered pair of numbers \( (x, y) \) corresponding to its coordinates: the first number \( x \) indicates the horizontal position, and the second number \( y \) indicates the vertical position. For instance, a point with coordinates \( (2, 3) \) would be found by moving 2 units to the right and 3 units upwards from the origin.

When plotting the function \( f(x) = 2^x \) in our exercise, we used the rectangular coordinate system to locate and plot five points corresponding to our selected x-values. This visual representation helps us understand the behavior of the function, showcasing its growth rate.

Vertical Shift of a Graph

A vertical shift of a graph is a transformation that moves every point of the original function up or down along the y-axis without altering its shape. This type of transformation can be represented algebraically by adding or subtracting a constant to the original function's output.

In the case of our functions \( f(x) = 2^x \) and \( g(x) = 2^x - 1 \), the subtraction of 1 from function \( f \) translates to a downward shift of the entire graph by one unit. This shift is why every y-value for the function \( g \) is exactly 1 less than the corresponding y-value for function \( f \).

To clearly visualize a vertical shift, first graph the original function and then apply the shift to each point. For instructional purposes, highlighting the original and shifted graphs in different colors or styles can aid in distinguishing the change.

Plotting Points in Algebra

Plotting points is a foundational skill in algebra that enables us to represent functions graphically. To effectively plot points for an exponential function like \( f(x) = 2^x \), we follow these steps:

• Determine a set of x-values to evaluate - typically, a range including both positive and negative integers is used.
• Calculate the corresponding y-values for each x-value by substituting them into the function's equation.
• Translate these x and y-value pairs into ordered pairs that correspond to points on the rectangular coordinate system.
• Mark these points on the graph and connect them appropriately to represent the function's curve.

When the x-values and their calculated y-values are plotted, the shape of the function starts to emerge. For both functions \( f(x) = 2^x \) and \( g(x) = 2^x - 1 \), plotting points helps illustrate the nature of exponential growth and the effect of transformations such as vertical shifts.