Problem 9 - Algebra and Trigonometry

Question

1–10 ? Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. (a) \(y=f(4 x) \quad\) (b) \(y=f\left(\frac{1}{4} x\right)\)

Step-by-Step Solution

Answer

(a) Compress horizontally by 4. (b) Stretch horizontally by 4.

Step 1: Understanding the Problem

We are given a function \( f(x) \) and we need to transform its graph based on two given functions: \( y=f(4x) \) and \( y=f\left(\frac{1}{4}x\right) \). We must describe how these transformations affect the original graph of \( f(x) \).

Step 1: Horizontal Compression

For the transformation \( y = f(4x) \), the factor \( 4 \) inside the function affects the graph horizontally. Specifically, it compresses the graph towards the y-axis by a factor of \( 4 \). This means that every x-coordinate on the graph is divided by 4.

Step 2: Horizontal Stretch

For the transformation \( y = f\left(\frac{1}{4} x\right) \), the factor \( \frac{1}{4} \) inside the function affects the graph horizontally by stretching it away from the y-axis. This means every x-coordinate on the graph is multiplied by 4.

Step 4: Conclusion

A horizontal compression by a factor of \( 4 \) is seen in \( y = f(4x) \) making the graph narrower, while a horizontal stretch by the same factor occurs in \( y = f\left(\frac{1}{4} x\right) \) making the graph wider.

Key Concepts

Horizontal Compression

A horizontal compression of a graph occurs when the graph is squeezed closer towards the y-axis. This happens for the function transformation \( y = f(4x) \). Here, the '4' inside the function is the key player. This factor compresses the graph of the original function horizontally by dividing each x-coordinate by 4.
Consider a simple example: if a point on the graph of \( f(x) \) had an x-coordinate of 8, after applying this transformation, it will move to \( \frac{8}{4} = 2 \).

• This compression makes the graph look narrower.
• Despite the squeezing, the overall shape of the graph doesn't change.
• Only the width changes due to the new positioning of points.

Remember, a horizontal compression does not affect the y-coordinates, only the x-coordinates are changed. This is special to horizontal transformations and is essential for solving problems related to graph transformations accurately.

Horizontal Stretch

In contrast to compression, a horizontal stretch causes the graph to expand away from the y-axis, making it look wider. The function transformation \( y = f\left(\frac{1}{4} x\right) \) illustrates this concept well.
Here, the factor \( \frac{1}{4} \) stretches the graph by multiplying every x-coordinate by 4. Imagine if a x-coordinate was 2 on the original graph of \( f(x) \); after stretching, it becomes \( 2 \times 4 = 8 \).

• The graph appears wider but maintains the same relationship between points on the curve.
• Only the distance between points on the x-axis changes, while the curve's form remains intact.
• The y-coordinates stay the same, similar to a compression transformation.

Remember, while the graph stretches horizontally, none of the vertical aspects are modified, ensuring the essence of the original graph remains preserved.

Function Transformations

Function transformations can seem complex at first, but breaking them down into basic movements makes them much clearer. A function transformation involves modifying a graph's appearance by altering its position, size, or shape.
In the context of horizontal transformations, we focus on changes caused by numbers multiplying the x-variable.

• Horizontal compression and stretching simply manipulate the width of a graph.
• These transformations come from multiplying the x-variable in the function by a constant.
• The aim is to maintain the original function's qualities while adjusting its layout.

Essentially, these transformations don't disrupt the graph's general shape, only how widely or narrowly it spans across the x-axis. This understanding is crucial for analyzing changes according to specific mathematical rules and learning how various parts of a function equation influence its graph.