Problem 13 - Algebra and Trigonometry

Question

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) \(\begin{array}{ll}{\text { (a) } y=f(4 x)} & {\text { (b) } y=f\left(\frac{1}{4} x\right)}\end{array}\)

Step-by-Step Solution

Answer

(a) The graph is horizontally compressed by a factor of 1/4. (b) The graph is horizontally stretched by a factor of 4.

Step 1: Understanding Horizontal Stretches and Compressions

In this exercise, we are transforming the graph of the function \(f(x)\) by modifying its input variable. This affects how the graph stretches or compresses horizontally. If the transformation involves multiplying the input \(x\) by a factor \(k\) in \(f(kx)\), then certain rules apply.

Step 2: Transformation Understanding for Part (a)

For the function \(y = f(4x)\), the input \(x\) is multiplied by 4. This results in a horizontal compression of the graph of \(f(x)\) by a factor of 1/4, since multiplying \(x\) by 4 actually speeds up the function's response to changes in \(x\). The graph is squeezed towards the y-axis.

Step 3: Transformation Understanding for Part (b)

For the function \(y = f\left(\frac{1}{4}x\right)\), the input \(x\) is multiplied by \(\frac{1}{4}\). This results in a horizontal stretch of the graph of \(f(x)\) by a factor of 4, expanding the graph away from the y-axis. This is because the function reacts slower to changes in \(x\), spreading out the graph.

Key Concepts

Horizontal Compression

When we talk about horizontal compression in the context of graph transformations, we’re discussing a change that affects the width of the graph. The most common way to compress a graph horizontally is by altering the input variable of a function. For example, if you have a function modified to be \(y = f(4x)\), this indicates that every \(x\)-value is multiplied by a factor of 4. However, the consequence is that the graph of the function is compressed towards the y-axis by a factor of \(\frac{1}{4}\).

• The input \(x\) is multiplied by a number greater than 1.
• The graph shrinks horizontally, bringing the points closer together.
• The key is to "perform the opposite": multiplying the input \(x\) results in compression by the reciprocal factor.

This adjustment essentially makes the function change more quickly over a smaller interval, presenting as a narrower graph spread.

Horizontal Stretch

Horizontal stretch widens the graph's spread along the x-axis. It is achieved by reducing the influences of the input variable in the function.A classic example can be seen with \(y = f\left(\frac{1}{4}x\right)\). By multiplying \(x\) by \(\frac{1}{4}\), what actually happens is a stretching of the graph horizontally by a factor of 4.

• The input \(x\) is multiplied by a number between 0 and 1.
• This causes the graph to stretch or widen horizontally.
• The mathematical operation "does the inverse": dividing the input space lengthens the graph by the reciprocal stretch factor.

Thus, the function changes more slowly over a larger interval, resulting in a more extensive curve.

Function Transformation

Function transformations are key to understanding how graphs behave under different manipulations. They enable us to predict and visualize changes to a graph's shape, position, and orientation. Specifically, transformations can include:

• Horizontal changes: Compression and stretch altering the width.
• Vertical changes: Shifting graphs up/down and stretching/compressing height.
• Reflections: Flipping the graph across axes in response to negative transformations.
• Translations: Moving the graph without altering its shape or size.

Understanding the specific effects of these manipulations helps us graph functions efficiently and with clarity.

Graphing Techniques

Graphing techniques are fundamental tools for visually representing functions and transformations. Mastery of these techniques can simplify complex function graphs and reveal underlying patterns or behaviors. Key techniques include:

• Identifying transformations: Recognize the type of transformation (e.g., stretch, compression).
• Sketching accurately: Use a consistent scale and correctly place key points.
• Using grid lines: Aid precision in sketches and identify intersections.
• Understanding symmetry and asymptotes: Predict behavior as graphs expand or contract.

Employing these techniques effectively can transform a seemingly complex function into an easily understandable visual aid.