Problem 55 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=4 f(x) $$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a vertical stretch of \(f(x)\) by a factor of 4.

Step 1: Identify the Original Function

The original function is given as \(f(x)\). We want to understand how the function \(g(x)\) is derived from \(f(x)\).

Step 2: Analyze the Transformation

The function \(g(x) = 4f(x)\) is derived by multiplying the original function \(f(x)\) by 4. This operation is a vertical stretch of \(f(x)\) by a factor of 4.

Step 3: Describe the Vertical Stretch

Multiplying \(f(x)\) by 4 means that the distance of each point on the graph from the x-axis is increased by 4 times, while the x-coordinates remain unchanged. This transformation does not affect the x-intercepts and only changes the y-values.

Key Concepts

Vertical Stretch

When we talk about a vertical stretch, we are discussing a specific type of function transformation. In the given exercise, we have the original function denoted as \( f(x) \), and the transformed function \( g(x) = 4f(x) \). As the name suggests, a vertical stretch involves enlarging, or stretching, the graph of the function vertically.
This happens by multiplying the y-values of \( f(x) \) by a constant factor, in this case, 4. Thus, every point \((x, y)\) on the graph of \( f(x) \) becomes \((x, 4y)\) on the graph of \( g(x) \).

• Each y-value is increased by fourfold.
• This makes the graph of \( g(x) \) appear taller compared to \( f(x) \).

It's important to note that vertical stretches do not affect the horizontal positioning—so the x-values remain the same. This type of transformation preserves the shape of the graph but alters its size in the vertical direction.

Original Function

The term original function typically refers to the base or starting function before any transformations are applied. In the context of this exercise, the original function is \( f(x) \). Understanding this function becomes essential because it serves as the benchmark or reference for assessing how the graph will change.
Before any transformations, \( f(x) \) has its distinct set of points, reflecting the relationship between x and y as described by its equation. Any transformation, like the vertical stretch given in the exercise, modifies this original setup in a specific manner.

• Knowing \( f(x) \) helps in predicting the effect of transformations.
• The graph's features like intercepts and turning points originate from these basic functions.

In many problems like this one, pinpointing the changes relative to the original function provides a clearer understanding of the transformations applied.

Graph of a Function

The graph of a function visually represents the function's behavior. It provides an intuitive understanding of how y-values change with x-values.
In our exercise, the task is to visualize \( g(x) = 4f(x) \) and how its graph is derived from the original function \( f(x) \).
Graphs are invaluable in mathematics because they make complex relationships easier to understand.

• They help in identifying function properties such as asymptotes, intercepts, and intervals of increase or decrease.
• Visual changes, like vertical stretches, become more apparent when analyzed through graph comparisons.

For any transformed function, examining the new graph will reveal the changes made. This can include variations in heights, widths, and positions based on the type of transformation applied.