Problem 35 - College Algebra

Question

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) 5\) units right

Step-by-Step Solution

Answer

The transformed function is \(f(x) = 4^{x-5}\).

Step 1: Understand the Transformation

The problem involves transforming the function \(f(x) = 4^x\). Specifically, it asks to shift \(f(x)\) 5 units to the right along the x-axis.

Step 2: Determine the Transformation Rule

To shift a function horizontally, use the transformation \(f(x - c)\), where \(c\) is the number of units to shift. Here, \(c = 5\) because we are shifting 5 units to the right.

Step 3: Apply the Transformation

Apply the transformation rule: replace \(x\) with \(x-5\) in the original function \(f(x) = 4^x\). This results in the new function \(f(x) = 4^{x-5}\).

Key Concepts

Function Transformations

In algebra, **function transformations** involve altering a function to produce a new function. These transformations can include shifting, scaling, reflecting, or rotating the graph of the original function.

Function transformations are significant because they help us understand how a function's shape can change when parameters are altered.

• Shifting: This involves moving the graph along the x-axis or y-axis.
• Scaling: This means stretching or compressing the graph vertically or horizontally.
• Reflecting: This is flipping the graph over a specific axis.
• Rotating: This is less common and typically occurs in more complex transformations.

The purpose of using transformations is to make the more complicated functions easier to solve by relating them back to simpler, known graphs. By adjusting the function, you can predict the behavior of the graph after a transformation is applied.

Horizontal Shift

When a function is shifted horizontally, every point on the graph of the function moves left or right by the same distance. This is called a **horizontal shift**.

A horizontal shift involves changing the function's input, or x-value, without affecting its output, or y-coordinates.

• If a function shifts to the right, we replace x with (x - c).
• If a function shifts to the left, we replace x with (x + c).

For instance, consider the function transformation of the exponential function \(f(x) = 4^x\). If we need to shift this function 5 units to the right, we would adjust the function to \(f(x) = 4^{x-5}\).
This transformation means that, before computing the power of 4, we first subtract 5 from any input value x.

Exponential Functions

**Exponential functions** are a type of mathematical function where a constant base is raised to a variable exponent. These functions are common in modeling growth and decay processes because they can describe rapid increases or decreases.

The general form of an exponential function is \(f(x) = a^x\), where:

• \(a\) is a constant base (greater than 0).
• \(x\) is the variable exponent.
• The value of \(f(x)\) represents the output.

For example, in the function \(f(x) = 4^x\), the base is 4. This tells us that as x increases, the function grows exponentially - meaning it increases much faster than a linear or quadratic function would.
Understanding how to manipulate these functions through transformations can help analyze and predict patterns in various real-life situations, such as population growth or radioactive decay.