Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired transformation. Do not use a calculator. The cubing function, vertically shrunk by applying a factor of \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{2}x^3 \).

Step 1: Understand the Original Function

The original function is the cubing function, which is given as \( y = x^3 \). This function is characterized by its input, \( x \), being raised to the third power.

Step 2: Apply the Vertical Shrink Transformation

A vertical shrink affects the outputs (\( y \)-values) by reducing their magnitude according to the shrink factor. Here, you need to apply a factor of \( \frac{1}{2} \). This means you multiply the entire output of the function by \( \frac{1}{2} \).

Step 3: Write the Transformed Equation

To apply the vertical shrink by a factor of \( \frac{1}{2} \), modify the equation of the original function to \( y = \frac{1}{2}x^3 \). This equation represents the vertically shrunk version of the cubing function.

Key Concepts

Understanding the Cubing Function

The cubing function is a fundamental mathematical relation expressed as \( y = x^3 \). This function takes each input \( x \) and raises it to the power of three. The output \( y \) represents the cube of \( x \). This function is crucial because it exhibits distinct properties that make it a part of polynomial functions. A typical graph of the cubing function shows a smooth curve passing through the origin and is symmetric about the origin. This symmetry is due to its odd nature, as it produces both positive and negative outputs based on whether \( x \) is positive or negative.

• The function grows quickly, producing large outputs for large inputs due to the cubing.
• Its graph is centered at the origin and extends infinitely in both directions.
• The curve passes through points like \((-1, -1)\), \((0, 0)\), and \((1, 1)\).

Exploring Vertical Shrink

Vertical shrink is a type of function transformation that alters how a graph behaves regarding the vertical axis. It compresses the graph towards the x-axis, effectively reducing the 'height' of all the outputs without altering the x-values. For the cubing function, to apply a vertical shrink, you multiply its entire output \( y \) by a specific factor. In this case, the vertical shrink factor is \( \frac{1}{2} \). This transformation means every point on the original graph moves closer to the x-axis by this factor.

• If a point was originally \( (x, y) \) on the graph of \( y = x^3 \), it becomes \( (x, \frac{1}{2}y) \) after shrinking.
• The graph maintains its original shape but becomes flatter.
• This kind of transformation is useful for visual adjustments in data modeling.

Grasping Function Transformation

Function transformation encompasses various techniques to adjust a function's graph for different outcomes. These transformations include translations, reflections, stretches, and shrinks. Each type serves specific purposes in functions' visualization and application. In the case of shrinking the cubing function, understanding how transformations affect graphs is vital. Vertically shrinking \( y = x^3 \) to \( y = \frac{1}{2}x^3 \) is an example of how factors can compress a function. Consider these elements:

• Transformation types include horizontal or vertical tweaks.
• Transformations help in modeling real-world scenarios by adjusting graphs to fit certain conditions.
• Knowing various transformations enables efficient function manipulation in mathematical problems.

Adjusting functions through these transformations can render them more suitable for solving equations or modeling situations we encounter mathematically and scientifically.