Problem 7 - A Graphical Approach to College Algebra

Question

Write the equation that results in the desired transformation. Do not use a calculator. The cubing function, vertically shrunk by applying a factor of 0.25 and reflected across the \(y\) -axis

Step-by-Step Solution

Answer

The equation is \( f(x) = -0.25x^3 \).

Step 1: Understand the Parent Function

The parent function given is the cubing function, which is represented by the equation \( f(x) = x^3 \).

Step 2: Apply the Vertical Shrink

A vertical shrink by a factor of 0.25 means you multiply the output (the entire function) by 0.25. Thus, the function becomes \( f(x) = 0.25x^3 \).

Step 3: Reflect Across the y-axis

To reflect a function across the \(y\)-axis, you replace \(x\) with \(-x\). So, substitute \(-x\) into the shrunk function: \( f(x) = 0.25(-x)^3 \).

Step 4: Simplify the Equation

Simplify \((-x)^3\) to \(-x^3\). Thus, the transformed function is \( f(x) = 0.25(-x^3) = -0.25x^3 \).

Key Concepts

Cubing Function

The cubing function is an essential aspect of mathematical functions and transformations. It serves as the foundation for many complex calculations and is represented by the equation \( f(x) = x^3 \). At its core, this function elevates the input value \(x\) to the power of three, producing a result that is always reflective of the cube's properties. The graph of a cubing function typically shows a distinctive S-shape, crossing through the origin. This symmetry around the origin ensures both positive and negative regions are equally represented.

• Cubing functions are defined for all real numbers \(x\).
• The growth of the function is quite rapid as \(x\) increases or decreases.
• This function is odd, meaning it maintains symmetry about the origin: \( f(-x) = -f(x) \).

Vertical Shrink

A vertical shrink is a type of transformation that compresses the function along the vertical axis. This occurs when each output \(f(x)\) of the function is multiplied by a factor between 0 and 1, effectively reducing the range of the graph. In the equation provided, \( f(x) = 0.25x^3 \), a vertical shrink by the factor of 0.25 results in each y-value being one-quarter of its original size. This transformation scales the graph towards the x-axis.

• Vertical shrink affects the y-coordinates, not altering the x-coordinates.
• It compresses the graph vertically, making it appear flatter.
• Ensures the graph keeps its original shape, only the amplitude changes.

Reflection Across the y-axis

Reflection across the y-axis modifies a graph by flipping it over the y-axis. To achieve this transformation, every x in the function is replaced with \(-x\). Applying this to the vertically shrunk function results in \( f(x) = 0.25(-x)^3 \). Since cube functions transform \( (-x)^3 \) into \( -x^3 \), this particularly drastic transformation results in a negation inside the function.

• This operation impacts the horizontal placement, not the shape.
• It reverses all x-values, effectively flipping the graph horizontally.
• A reflection retains the original dimension proportions but swaps sides.

Parent Function

In the realm of function transformations, the term "parent function" refers to the simplest form of a given family of functions. For the cubic family, the parent function is \(f(x) = x^3\). This base function acts as the cornerstone from which transformations like translations, reflections, vertical stretches, and shrinks are applied. Understanding the parent function is essential as it provides insights into how transformations alter the graph's appearance while retaining the fundamental properties of the function type.

• Parent functions are inherently simple, forming the root of function families.
• Alterations of parent functions result in recognizable transformations.
• Understanding parent functions aids in predicting outcomes of transformations.