Problem 103 - Precalculus

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 \( g(x) = 4^{x-5} \).

Step 1: Identify the Base Function

The given function is \( f(x) = 4^x \). This is an exponential function with a base of 4.

Step 2: Understand the Transformation

The problem asks for the function to be shifted 5 units to the right. This kind of shift is a horizontal translation of the function.

Step 3: Apply the Horizontal Shift

To shift the function \( f(x) = 4^x \) 5 units to the right, we replace \( x \) with \( x-5 \). The transformed function becomes \( g(x) = 4^{x-5} \).

Key Concepts

Horizontal Translation

In mathematics, a horizontal translation is a type of transformation that shifts a graph left or right along the x-axis. For any given function, when you want to move it horizontally, you adjust the variable within the function. Here's how it works for our exponential function.

• To shift right, you subtract from the x-value: use \( x - c \) in place of \( x \), where \( c \) is the number of units to shift right.
• To shift left, you add to the x-value: use \( x + c \) in place of \( x \), with \( c \) units shifted left.

In our exercise, the function \( f(x) = 4^x \) is shifted 5 units to the right. This results in substituting \( x \) with \( x-5 \), thus forming \( g(x) = 4^{x-5} \). The graph of the function moves entirely to the right without changing its shape. This transformation is quite prevalent in graph manipulation, making understanding its effects crucial when studying functions.

Graph of Functions

The graph of a function is a visual representation of all the points \((x, y)\) that satisfy the function. For exponential functions such as \( f(x) = 4^x \), the graph will typically have some distinctive characteristics:

• It rises steeply, demonstrating exponential growth.
• The graph passes through the point (0, 1) because \( 4^0 = 1 \).
• As \( x \) becomes negative, the graph approaches the x-axis (asymptote) but never touches it.

Understanding the basics of plotting such functions aids in visualizing how these functions behave. When applying transformations like horizontal shifts, it's important to understand that the essential shape of the graph won't change, it will simply be relocated along the x-axis as per the transformation rules.

Exponential Function Properties

Exponential functions are powerful mathematical functions with unique properties:

• They have a constant base raised to a variable exponent, like \( 4^x \) in our example.
• The domain of these functions is all real numbers, meaning they accept any x-value.
• Their range is always positive, never touching or falling below the x-axis.

Such functions exhibit a swift increase or decrease, which defines their exponential nature. This property makes exponential functions ideal for modeling real-world phenomena involving growth or decay, such as population growth, radioactive decay, and financial interest calculations. When transforming these functions graphically, remembering these properties ensures accurate interpretation and manipulation of the exponents.