Problem 4 - A Graphical Approach to College Algebra

Question

Write the equation that results in the desired transformation. Do not use a calculator. The cube root function, reflected across the \(x\) -axis

Step-by-Step Solution

Answer

The equation is \(-\sqrt[3]{x}\).

Step 1: Identify the Parent Function

The parent function for a cube root is \( f(x) = \sqrt[3]{x} \). This is the original function we will reflect across the \(x\)-axis.

Step 2: Understand Reflection Across the x-axis

Reflecting a function across the \(x\)-axis involves multiplying the function by \(-1\). This changes the sign of the output values but not the input.

Step 3: Apply the Reflection Transformation

To reflect \( f(x) = \sqrt[3]{x} \) across the \(x\)-axis, multiply the entire function by \(-1\). Thus, the new function becomes \( -\sqrt[3]{x} \).

Key Concepts

Reflection Across the x-axis

Reflection across the x-axis is a type of function transformation that flips a graph over the horizontal axis. This transformation changes the sign of the output (dependent variable, typically y) while leaving the input (independent variable, typically x) unchanged.
When a function like \( f(x) = \sqrt[3]{x} \) undergoes reflection, the entire expression is multiplied by \(-1\).
This means each point on the graph now mirrors directly below its original position. For example, if the original cube root function has a point at (2, 1.26), the reflected function will show a point at (2, -1.26).

• This swapping is due to the change from \( f(x) \) to \( -f(x) \).
• This gives an inverted appearance about the x-axis and doesn't affect the horizontal location of points.

Cube Root Function

The cube root function is a vital concept in function transformations and is denoted by \( f(x) = \sqrt[3]{x} \). It describes the inverse operation of cubing a number. Unlike the square root function, which is only defined for non-negative inputs due to real numbers, the cube root function accepts all real numbers, both positive and negative.
This results in a function with a characteristic "S" shape curve, passing through the origin (0,0). Important features of the cube root graph include its horizontal symmetry about the origin. It increases slowly for negative x-values, passes through the origin, and continues growing for positive x-values.

• The cube root function's domain and range are both all real numbers (\(-\infty, \infty\)).
• The function is odd, showcasing symmetry with respect to the origin: \( f(-x) = -f(x) \).

Understanding the cube root function is essential when performing transformations like reflections across axes.

Parent Function

A parent function is the simplest form of a set of functions that form a family. It serves as the foundational template from which more complex functions are derived through transformations.
For the cube root function family, the parent function is \( f(x) = \sqrt[3]{x} \). This base function has not undergone any transformations, such as shifting, stretching, or reflecting. Instead, it provides a benchmark for all modifications and comparisons.
Common transformations applied to parent functions include:

• Reflections, which flip the graph across an axis (as with the x-axis reflection).
• Translations, which shift the graph horizontally or vertically.
• Dilations, which stretch or compress the graph.

Understanding parent functions aids in grasping later transformations by providing a clear comparison point.