Problem 35 - College Algebra with Corequisite Support

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 \(f(x) \quad 5\) units right

Step-by-Step Solution

Answer

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

Step 1: Understand the Transformation

The given function is \(f(x) = 4^x\). A horizontal shift to the right involves modifying the input \(x\) of the function. Specifically, a shift to the right by \(c\) units is achieved by replacing \(x\) with \(x - c\). In this case, \(c = 5\).

Step 2: Apply the Horizontal Shift

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

Step 3: Write the Transformed Function

The function after applying the transformation is \(g(x) = 4^{x-5}\). This represents the graph of \(f(x)\) shifted 5 units to the right.

Key Concepts

Horizontal Shift

To understand what a horizontal shift is, consider an imaginary line where your graph normally resides. By shifting this line to the right or left, you're performing a horizontal shift.
A horizontal shift affects the input value of a function. When you’re told to shift a function horizontally, you’ll either increase or decrease the input (usually represented as "x") by a certain number of units.

• **Right Shift**: When a function is shifted to the right by "c" units, you subtract "c" from "x". This results in the transformation: replace "x" with "x - c". For example, shifting the function to the right by 5 units changes it to: \(f(x - 5)\).
• **Left Shift**: Conversely, for a shift to the left, you add "c" to "x", resulting in replacing "x" with "x + c".

Horizontal shifts are straightforward when visualized: picture picking up the graph and sliding it sideways on the x-axis without altering its shape."
Understanding how this concept works is key for mastering graph transformations.

Exponential Functions

Exponential functions form a crucial part of algebra and calculus. They are defined by the equation \(f(x) = a^x\), where "a" is a constant. They are unique because they grow or decay at an ever-accelerating rate due to the variable "x" being in the exponent. In our example, the function \(f(x) = 4^x\) is an exponential function, which means that for each unit increase in "x", the function's output gets multiplied by the base, which is 4 in this case.

• **Base Greater Than 1**: Results in exponential growth, making the graph rise sharply.
• **Basic Characteristics**: The initial value is at \(f(0) = 1\) if "a" is normal. The graph will keep approaching 0 as "x" approaches negative infinity, representing a horizontal asymptote at \(y = 0\).

Exponential functions are prevalent in many real-world situations where things grow or shrink rapidly, such as populations, radioactive decay, or financial investments.

Graph Transformations

Graph transformations allow us to take a basic function and manipulate its graph in various ways to create a new one. These transformations include shifts, reflections, stretches, and compressions.
This flexibility is not only mathematical but offers insights into how real-world phenomena can be modeled by altering simple equations.

• **Vertical Shifts and Stretches**: Adjust along the y-axis. Moving graphs up or down, or making them steeper.
• **Reflection and Compression**: Changes reflect graphs in axes or transform their wideness. Both vertical and horizontal reflections are possible.

To apply these correctly, pay attention to how the function is adjusted. Besides horizontal shifts like in this exercise, other transformations might add or multiply constants to the function directly. These transformation techniques help in predicting future patterns or analyzing historical data based on new conditions.