Problem 35 - 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. $$f(x)=2^{x} \text { and } g(x)=2^{x+1}-1$$

Step-by-Step Solution

Answer

The function \(g(x)\) is a one-unit vertical shift down from the function \(f(x)\).

Step 1: Graph Function \(f\)

First, calculate \(f(x)\) for each integer value from -2 to 2. The results are: \(-2 \rightarrow 2^{-2}=0.25\), \(-1 \rightarrow 2^{-1}=0.5\), \(0 \rightarrow 2^{0}=1\), \(1 \rightarrow 2^{1}=2\), \(2 \rightarrow 2^{2}=4\). Now, plot these points on a graph.

Step 2: Graph Function \(g\)

Next, calculate \(g(x)\) for each integer value from -2 to 2. The results are: \(-2 \rightarrow 2^{-2+1}-1 = 2^{-1}-1 = 0.5-1 = -0.5\), \(-1 \rightarrow 2^{-1+1}-1 = 2^{0}-1 = 1-1 = 0\), \(0 \rightarrow 2^{0+1}-1 = 2^{1}-1 = 2-1 = 1\), \(1 \rightarrow 2^{1+1}-1 = 2^{2}-1 = 4-1 = 3\), \(2 \rightarrow 2^{2+1}-1 = 2^{3}-1 = 8-1 = 7\). Plot these points on the same graph as \(f(x)\).

Step 3: Analyze the Graphs

Examine the graphs of \(f\) and \(g\). It can be observed that for every given \(x\), \(g(x)\) is exactly one unit down from \(f(x)\). This means \(g(x)\) is a vertical shift of \(f(x)\) down by 1 unit.

Step 4: Confirmation using graphing utility

To confirm this observation, use a graphing utility and plot the graphs of both the functions. The utility will also show that \(g\) is a 1 unit vertical shift down from \(f\).

Key Concepts

Coordinate System

Understanding the coordinate system is the first step when working with graphs. A rectangular coordinate system, also known as the Cartesian coordinate system, consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
These axes divide the plane into four quadrants, making it easy to locate points using pairs of numbers known as coordinates. For example, the point \((2,3)\) represents a position 2 units along the x-axis and 3 units up the y-axis.

• The x-axis is where you place values for the input of your function, also called the independent variable.
• The y-axis measures the output or result of the function, known as the dependent variable.

When graphing functions like \(f(x) = 2^x\) and \(g(x) = 2^{x+1} - 1\), you need to plot the output for each x-value and connect these points to visualize the function.

Function Transformation

Function transformation is a powerful technique in math used to alter the graph of a function. It involves shifting, stretching, compressing, or flipping the graph of a function in various ways.
In the case of \(g(x) = 2^{x+1} - 1\), we can observe a two-step transformation from \(f(x) = 2^x\):

• Horizontal Shift: The \(x+1\) inside the exponent suggests a shift left. Every point on the graph of \(f\) moves one unit to the left.
• Vertical Shift: The \(-1\) means a downward shift. Therefore, every point on the graph of \(f(x)\) is moved one unit down to create \(g(x)\).

These transformations help understand how functions relate to each other and establish patterns between their graphs without having to plot each point individually.

Exponential Functions

Exponential functions are a class of mathematical functions where a constant base is raised to a variable exponent. They are expressed in the form \(f(x) = a^x\), where \(a\) is a positive constant. These functions are known for their distinctive curved graphs that either grow rapidly (exponential growth) or decrease quickly (exponential decay).

• The function \(f(x) = 2^x\) is an example of an exponential growth function, where the value of \(f(x)\) doubles as \(x\) increases by 1.
• For exponential functions, the y-intercept is typically at \((0,1)\) if no transformations are applied.

In practice, exponential functions model real-life situations like population growth and radioactive decay, making understanding their behavior essential in various fields. The steepness and direction of their graphs can also be altered by transformations, as seen in the comparative shift from \(f(x)\) to \(g(x)\).