Question

Suppose that the graph of a function \(f\) is known. Explain how the graph of \(y=4 f(x)\) differs from the graph of \(y=f(4 x)\).

Short answer

The graph of \(y = 4f(x)\) is vertically stretched by 4, while the graph of \(y = f(4x)\) is horizontally compressed by 4.

Step-by-step solution

Identify the original function

Start by understanding the original function, which is given as \(y = f(x)\). This is the base graph we will use for comparison.

Understand the transformation for \(y = 4f(x)\)

The transformation \(y = 4f(x)\) involves multiplying the output (or \(y\)-value) of \(f(x)\) by 4. This means the graph of \(y = f(x)\) will be vertically stretched by a factor of 4. For any \(x\), if \(f(x) = y\), then \(4f(x) = 4y\). Thus, all points on the graph of \(y = f(x)\) are shifted vertically to \(4\) times their original height.

Understand the transformation for \(y = f(4x)\)

The transformation \(y = f(4x)\) involves multiplying the input (or \(x\)-value) of \(f(x)\) by 4 before applying the function. This results in a horizontal compression of the graph by a factor of \(1/4\). For any \(x\), if \(f(x) = y\), then \(f(4x) = y\). Hence, the graph is compressed along the \(x\)-axis, making it appear four times narrower.

Compare the two transformations

Comparing the two transformations: \(y = 4f(x)\) stretches the graph of \(y = f(x)\) vertically by a factor of 4, while \(y = f(4x)\) compresses the graph horizontally by a factor of 4. The first affects the \(y\)-values, scaling them to four times their original height, while the second affects the \(x\)-values, scaling them to a quarter of their original width.

Key concepts

vertical stretch

A vertical stretch is when the graph of a function is stretched vertically. Imagine pulling a rubber band from the top to make it taller. This happens when you multiply the entire function by a number greater than 1.
For example, in the transformation \(y = 4f(x)\), if the original output of the function \(f(x)\) was a particular value \(y\), this value is multiplied by 4.

• If \(f(x) = 1\), then \(4f(x) = 4\).
• If \(f(x) = -2\), then \(4f(x) = -8\).

This means every point on the graph moves up or down, making the graph appear taller. Remember:
- A vertical stretch keeps the shape of the graph but changes its height.
- It affects the y-values, not the x-values.

horizontal compression

A horizontal compression happens when the graph of a function is squashed horizontally. Think of compressing a spring sideways to make it narrower.
This occurs when you multiply the input, or \(x\)-value, by a number greater than 1 before applying the function.
For example, in the transformation \(y = f(4x)\), if the original function \(f(x)\) provided an output \(y\) for an input \(x\), now the function takes \(4x\) as the input.

• If \(f(1) = y\), then \(f(4 \times 1) = y \)
• If \(f(3) = y\), then \(f(4 \times 3) = y\)

Hence, each point on the graph moves horizontally towards the y-axis, making the graph look narrower by a factor of 4. Keep in mind:
- A horizontal compression changes the width of the graph but retains its overall shape.
- It affects the x-values, not the y-values.

graph transformations

Graph transformations change the position, shape, and orientation of a graph. They help in understanding how alterations in the equation impact the graph's appearance.

• Vertical Stretches and Compressions: Changing the y-values of the function by multiplying them alters their height.
  Example: \(y = c \, f(x)\), where \(c > 1\)


• Horizontal Stretches and Compressions: Modifying the x-values by multiplying inside the function changes their width.
  Example: \(y = f(bx)\), where \(b > 1\)

To practically compare:

• For \(y = 4f(x)\): Multiply \(y\) values by 4 (vertical stretch).
• For \(y = f(4x)\): Multiply \(x\) values by 1/4 (horizontal compression).

Understanding these transformations is key to mastering function graphing. They directly show how graph shapes shift with changes in their equations.