Problem 4 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

In the case of \(c > 1\), \(h(x) = f(cx)\) represents a horizontal compression. Conversely, for \(0 < c < 1\), it represents a horizontal stretch.

Step 1: Understand the given function

Start by recognizing that \(h(x) = f(cx)\) is a change of scale transformation on the function \(f(x)\). In this transformation, the input \(x\) has been scaled by a factor of \(c\), which will affect the horizontal spacing or stretch of the function.

Step 2: Evaluate for \(c > 1\)

Approach the \(c > 1\) case. Considering \(c\) as a scaling factor, a larger \(c\) (i.e., \(c > 1\)) would compress the function horizontally, making it narrower, compared to the original function. This transformation is identified as a horizontal compression.

Step 3: Evaluate for \(0 < c < 1\)

Next, consider the \(0 < c < 1\) case. A smaller \(c\) (i.e., \(0 < c < 1\)) would stretch the function horizontally, making it wider in comparison to the original function. This kind of transformation is called a horizontal stretch.

Key Concepts

Horizontal Compression

Horizontal compression occurs in function transformations when the input variable is scaled by a factor greater than one. Imagine shrinking something horizontally. This means that all the points that make up the graph of the function come closer together.When we apply a transformation represented by the formula \[ h(x) = f(cx) \] and the value of \( c \) is greater than 1, we're effectively making the graph narrower. Think of it as squeezing the function from the sides towards the y-axis.

• For example, if \( c = 2 \) in \( h(x) = f(2x) \), each point on the graph of \( h(x) \) is half as far from the y-axis compared to the same point on \( f(x) \).
• This means that the graph of the function will look steeper and more compressed.

Horizontal compressions help us understand how changing the input affects the width of the graph. It's all about how tightly the graph can fit along the x-axis in a squeezed manner.

Horizontal Stretch

A horizontal stretch in function transformations is quite the opposite of horizontal compression. In this case, you are extending the graph horizontally.With the transformation \[ h(x) = f(cx) \] if \( c \) is a value between 0 and 1, the graph of the function stretches out horizontally. This makes it appear wider compared to the original.

• For instance, if \( c = 0.5 \), every x-value for \( h(x) = f(0.5x) \) is doubled compared to the original function \( f(x) \).
• This essentially pulls the graph away from the y-axis, making it look flatter and more extensive.

Understanding horizontal stretch helps us envision how the graph can broaden across the x-axis, giving us different perspectives on how functions behave under different transformations.

Function Transformations

Function transformations involve modifying a function in a way that changes its shape or position on the graph. The formula \[ h(x) = f(cx) \] is a fundamental example of nonrigid transformations, which specifically change the horizontal characteristics of a function.These transformations impact functions in several ways:

• Scaling the input affects how the graph stretches or compresses.
• The parameter \( c \) controls whether a function gets stretched (\( 0 < c < 1\)) or compressed (\( c > 1\)).
• Function transformations are crucial for modeling real-world situations where scaling is involved.

It's all about recognizing how these transformations alter the representation of a function, enabling a better grasp of the subtleties involved in mathematical relationships.