Question

Match the function with the correct transformation of the graph of \(f\). Explain your reasoning. \(y=f(2 x)\)

Short answer

The transformation \(f(2x)\) compresses the graph of the function \(f(x)\) horizontally by a factor of 1/2.

Step-by-step solution

Recognize the transformation type

The transformation here is \(f(2x)\). This transformation form \(f(kx)\) represents a horizontal compression or stretch by a factor of 1/k. If k > 1, the graph of the parent function is compressed horizontally by a factor of 1/k.

Apply the transformation

In this case, k=2, therefore the graph of \(f(x)\) is compressed horizontally by a factor of 1/2. This means that the x-values are half of what they would be in the original function \(f(x)\). It 'speeds up' the function.

Key concepts

horizontal compression

A horizontal compression is a transformation that alters the horizontal span of a graph. Consider a function transformation of the form \( f(kx) \). When \(k > 1\), the function undergoes a horizontal compression. This effectively reduces the distance between the points on the graph horizontally.

The fundamental idea behind horizontal compression is that it "squeezes" the graph towards the y-axis. In simpler terms, it makes the graph narrower. For instance, in the transformation \( f(2x) \), each x-value of the parent function has to be divided by 2.

This results in each point moving closer to the y-axis by half its original distance. Therefore, every point on the graph will appear to move 'faster' as it reaches the same y-values in shorter horizontal distances.

graph transformations

Graph transformations adjust the appearance or position of a graph in various ways without changing its overall shape. These transformations can be categorized into several types, including translations, reflections, stretches, and compressions.

• Translations move the graph horizontally or vertically, depending on the function transformation applied.
• Reflections flip the graph over an axis, usually resulting in a mirror image of the original graph.
• Stretches and compressions affect the scale of the graph, either vertically or horizontally.

The key purpose of graph transformations is to modify the graph's position or scale without altering its essential characteristics. Understanding how each transformation works helps you manipulate graphs effectively for various applications.

parent function

A parent function is the simplest form of a function in a family. It's the base function from which transformations and alterations are made. For example, the parent function \(f(x) = x^2\) represents a simple parabola.

When comparing graph transformations, especially horizontal compressions or stretches, it's crucial to recognize the parent function. This is because understanding the fundamental shape or behavior of the parent function allows you to predict how changes like \(f(2x)\) will move or reshape the graph.

The parent function serves as the "original" or "reference" graph before any transformation. Knowing your parent function well means you can quickly spot transformations and predict how they'll look graphically.