Problem 27 - College Algebra

Question

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{x}-1\)

Step-by-Step Solution

Answer

The graph of \(g(x)=2^{x}-1\) is the graph of \(f(x)=2^{x}\) shifted downward by one unit.

Step 1: Graph the initial function

Start by graphing the base function \(f(x)=2^{x}\). For this function, as \(x\) increases, \(f(x)\) increases exponentially.

Step 2: Understand the transformation

The function \(g(x)\) can be obtained by downward shifting the graph of \(f(x) = 2^{x}\) by 1 unit. This is represented by the term \(-1\) in \(g(x) = 2^{x} - 1\). In general, if the function \(f(x)\) is transformed to \(f(x) - c\), where \(c\) is a positive constant, it represents a vertical shift downwards by \(c\) units.

Step 3: Graph the transformed function

To graph \(g(x) = 2^{x} - 1\), apply the downward shift to each point of the \(f(x)=2^{x}\) graph. This means subtracting one from the \(y\)-coordinate of each point in the graph of \(f(x)\). Plot these new points to create the graph for \(g(x)\).

Step 4: Confirm with a graphing utility

Draw these graphs using a graphing utility to confirm the hand-drawings. The graph of \(g(x)\) should be identical to the graph of \(f(x)\), but shifted one unit down across all \(x\)-values.

Key Concepts

Exponential Functions

Exponential functions are a key mathematical concept where the variable is an exponent. In the function \( f(x) = 2^x \), the base, 2, is a constant while \( x \) is the variable in the exponent. This type of function is a powerful tool in modeling various real-world scenarios, such as compound interest and population growth, where values increase rapidly.

Understanding the graph of an exponential function like \( f(x) = 2^x \) involves observing how quickly it rises as \( x \) increases. Starting from low values near zero, the graph shows that every increment in \( x \) leads to a doubling effect in the function's value. Initially, this growth might seem slow, but it quickly escalates due to the nature of exponentiation.

Key characteristics of the graph of \( f(x) = 2^x \):

• It passes through the point (0, 1) because any non-zero number to the power of zero is 1.
• The graph is asymptotic to the x-axis, meaning it gets infinitely close but never touches or crosses it.
• It is always increasing and never decreasing, displaying exponential growth as \( x \) becomes larger.

Vertical Shift

A vertical shift in graph transformations refers to moving the entire graph of a function up or down along the y-axis without altering its shape. In the given exercise, the function \( g(x) = 2^x - 1 \) embodies a vertical shift of the function \( f(x) = 2^x \) downward by one unit. This transformation is straightforward yet essential in adapting functions to meet problem-solving needs.

When a function \( f(x) \) is altered to become \( f(x) - c \), where \( c \) is a positive constant, each point on the graph of \( f(x) \) is shifted downward by \( c \) units. More generally:

• If \( c \) is positive, the graph shifts down by \( c \) units.
• If \( c \) is negative, the graph shifts up by \(|c|\) units.

In the example of \( g(x) = 2^x - 1 \), subtracting one from the function \( f(x) = 2^x \) lowers every point by one unit on the y-axis. This adjustment means that instead of approaching a minimum y-value of zero, \( g(x) \) approaches \(-1\) instead.

Graphing Utility

Graphing utilities or tools are an excellent way to visualize mathematical functions and confirm transformations like the ones discussed here. They can range from graphing calculators to online graphing software, each providing different capabilities for exploring and studying mathematical graphs.

When dealing with transformations such as vertical shifts, a graphing utility is invaluable. It enables you to:

• Quickly plot both the original function and its transformed version to see the effects of changes like shifts.
• 
  Identify intersection points and asymptotic behavior, ensuring a clearer understanding of how functions behave across various domains.
• Adjust the view window to focus on critical parts of the graph.

In practice, using a graphing utility allows you to observe the expected behavior of \( g(x) = 2^x - 1 \), verifying the one-unit downward shift visually. Therefore, these tools are crucial in both educational settings and practical applications, helping users to corroborate their theoretical calculations with visual confirmations.