Problem 5 - Precalculus with Limits

Question

A nonrigid transformation of \(y = f(x)\) represented by \(g(x) = cf(x)\) is a ________ ________ if \(c > 1\) and a ________ ________ if \(0 < c < 1\).

Step-by-Step Solution

Answer

The nonrigid transformation of \(y = f(x)\) represented by \(g(x) = cf(x)\) is a vertical stretch if \(c > 1\) and a vertical compression if \(0 < c < 1\).

Step 1: Analyze the value of the constant c greater than 1

The original function \(y = f(x)\) is transformed into \(g(x) = cf(x)\) where \(c > 1\). This means that for each \(x\), \(g(x)\) is \(c\) times larger than \(f(x)\). This stretches the graph of \(f(x)\) vertically.

Step 2: Identify the type of transformation for c > 1

For \(c > 1\), the function \(y = f(x)\) is vertically stretched. So when the constant \(c\) is greater than 1 in the transformation \(g(x) = cf(x)\), it is a vertical stretch.

Step 3: Analyze the value of the constant c between 0 and 1

The original function \(y = f(x)\) is transformed into \(g(x) = cf(x)\) where \(0 < c < 1\). This means that for each \(x\), \(g(x)\) is 'c' times smaller than \(f(x)\). This compresses the graph of \(f(x)\) vertically.

Step 4: Identify the type of transformation for 0 < c < 1

For \(0 < c < 1\), the function \(y = f(x)\) is vertically compressed. So when the constant \(c\) is between 0 and 1 in the transformation \(g(x) = cf(x)\), it is a vertical compression.

Key Concepts

Vertical Stretch

A vertical stretch happens when we take a function and transform it by multiplying it with a constant greater than one. If the original function is represented as \( y = f(x) \), the transformed function becomes \( g(x) = cf(x) \) where \( c > 1 \). This means that for every input \( x \), the output \( g(x) \) is multiplied by \( c \).
This transformation causes the graph of the function to stretch away from the x-axis. Imagine pulling a graph upward and downward; that's what a vertical stretch does.

• It effectively makes the function's values larger.
• Each point on the graph gets pushed farther from the x-axis.

Understanding vertical stretches is crucial because they affect the rate at which the function grows, making peaks higher and troughs deeper. If you're looking at a real-life situation modeled by a function, a vertical stretch could mean amplified changes at different points.

Vertical Compression

Contrary to a vertical stretch, a vertical compression occurs when you multiply the function by a constant less than one. If your original function is \( y = f(x) \), transforming it with \( g(x) = cf(x) \) where \( 0 < c < 1 \) results in a vertical compression. Here, \( g(x) \) becomes a fraction of the original \( f(x) \) at each \( x \).
This transformation 'squashes' the graph towards the x-axis. Imagine gently pressing down on the graph so it flattens out.

• Function values decrease, shrinking towards the axis.
• Peaks and valleys become less pronounced.

Vertical compression plays a big role in making functions less steep. It alters the dynamics of the function's behavior, which can be useful when you want a more restrained or moderated representation of data or mathematical models.

Function Transformation

Function transformation refers to any change or manipulation applied to a function's graph. It includes operations like vertical stretches, compressions, reflections, and shifts. The notion of transforming functions is a central aspect of studying graphs because it helps in visualizing and analyzing various real-world situations.
When dealing with transformations, it's important to focus on how specific changes to the function, like scaling or translating, influence the appearance of its graph. Here are the different ways transformations can affect a function:

• Scaling vertically or horizontally (stretch or compression).
• Shifting graphs up, down, left, or right.
• Reflecting graphs across axes.

By understanding transformations, you learn how to manipulate graphs to suit your needs, making predictions, and solving problems becomes more intuitive. Whether you're dealing with a complex physical system or a simple algebraic equation, mastering transformations allows you to see the connections and gain deeper insights into the behavior of functions.