Problem 33 - 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 \(f(x) 3\) units downward

Step-by-Step Solution

Answer

The function is \( g(x) = 4^x - 3 \).

Step 1: Identify the Original Function

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

Step 2: Understand the Transformation

The problem asks us to shift the graph of \( f(x) \) 3 units downward. In function transformations, shifting a graph downward involves subtracting a constant value from the function.

Step 3: Apply the Transformation

To shift \( f(x) = 4^x \) downward by 3 units, subtract 3 from the entire function. This results in the transformed function \( g(x) = 4^x - 3 \).

Step 4: Write the New Function

The function that represents the graph of \( f(x) = 4^x \) shifted 3 units downward is \( g(x) = 4^x - 3 \). This accounts for the vertical translation downward.

Key Concepts

Function Transformations

Function transformations allow us to shift, stretch, compress, or reflect a graph of a function. These transformations change the appearance and position of the function's graph but not its fundamental nature. For example, when transforming a function like an exponential one, such as \(f(x) = 4^x\), you can apply different types of transformations to move the graph in various ways.
Transformations can be classified as follows:

• **Translation**: Moves the graph without changing its overall shape. This includes vertical and horizontal shifts.
• **Reflection**: Flips the graph over a specified axis.
• **Stretching and Compressing**: Alters the graph's steepness by stretching or compressing it vertically or horizontally.

In many cases, students find it helpful to first apply translations and then other transformations, as translations are relatively straightforward compared to stretching or reflecting.

Vertical Translation

A vertical translation is a type of function transformation that moves the graph up or down. This is achieved by adding or subtracting a constant from the function. If the graph of a function is moved upwards, a positive constant is added. Conversely, if it's shifted downwards, a negative constant is subtracted.
For instance, with the exponential function \(f(x) = 4^x\), a vertical translation involves modifying the output value of the function. If you move \(f(x)\) by \(c\) units vertically, this results in:

• **Upward Shift**: \(h(x) = 4^x + c\)
• **Downward Shift**: \(g(x) = 4^x - c\)

The function maintains the same base, but the graph is relocated either up or down on the coordinate plane.

Downward Shift

The downward shift is a specific kind of vertical translation where the graph of a function is moved lower on the y-axis. This transformation involves subtracting a positive constant from the function's formula.
Consider our example, \(f(x) = 4^x\), which we shifted down by 3 units. The new function \(g(x) = 4^x - 3\) represents this transformation. Here, "-3" indicates that every y-value of the original function is now 3 units lower than before.

• **Impact**: The overall shape of the function remains unchanged. It continues to increase exponentially. However, its y-intercept will be lower.
• **Visualization**: Imagine the original graph being "pushed down" by 3 units. The entire graph retains its shape but aligns with new y-axis values.

This transformation is crucial in shifts because it enables finer adjustments of function placements without altering their fundamental growth or decay rates.