Question

Write the function whose graph is the graph of \(y=x^{3},\) but is: Reflected about the \(y\) -axis

Short answer

The function is \(y = -x^3\).

Step-by-step solution

- Understand the basic function

The given function is \(y = x^3\). This is the basic cubic function.

- Identify the transformation

The problem requires reflecting the graph about the \(y\)-axis. A reflection about the \(y\)-axis changes \(x\) to \(-x\).

- Apply the transformation

To reflect the function \(y = x^3\) about the \(y\)-axis, replace every \(x\) with \(-x\). The new function becomes \(y = (-x)^3\).

- Simplify the function

Simplify the function \((-x)^3\). Since \(-x\) raised to an odd power results in \(-x^3\), the function simplifies to \(y = -x^3\).

- Write the final function

The function whose graph is the reflection of \(y = x^3\) about the \(y\)-axis is \(y = -x^3\).

Key concepts

Cubic Function

A cubic function is a polynomial function of degree three, and the standard form is given by the equation \(y = x^3\). This type of function is characterized by an S-shaped curve, which can take different forms depending on its coefficients.
In our example, we are dealing with the simplest form of the cubic function, where \(y = x^3\). This graph passes through the origin (0, 0) and extends indefinitely in both positive and negative directions for both \(x\) and \(y\). Here are a few properties to keep in mind:

• The cubic function is symmetric about the origin.
• As \(x\) increases, \(y\) also increases and vice versa.
• As \(x\) decreases, \(y\) also decreases.

Understanding this basic property is crucial before applying any transformations.

Reflection

Reflection is a type of transformation that 'flips' a graph over a specific line. In this exercise, we reflect the graph of \(y = x^3\) about the \(y\)-axis.
When reflecting a graph over the \(y\)-axis, each \(x\)-coordinate of the original function is replaced by its negative counterpart. Let's break it down:

• If you have a point (a, b) on the original graph, after reflection it becomes (-a, b).
  

So, to reflect \(y = x^3\) about the \(y\)-axis, we change every instance of \(x\) to \(-x\). Following this step, our new function becomes \(y = (-x)^3\). When we simplify \((-x)^3\), we get \(y = -x^3\) because raising a negative number to the power of three results in a negative number. In summary, the reflection of the cubic function \(y = x^3\) about the \(y\)-axis is \(y = -x^3\). This change allows us to observe how the graph 'flips' to the left side while its basic S-shape remains unchanged.

Coordinate Transformation

Coordinate transformation involves changing the position or orientation of a graph on the coordinate plane without altering its shape. Reflecting, stretching, compressing, shifting, and rotating are various types of transformations.
Let's focus specifically on coordinate reflection, which is part of our exercise. Reflection over the \(y\)-axis is one of the simplest examples of coordinate transformation. This type of transformation modifies the \(x\)-component of the coordinates:

• Every x-value of the original coordinate turns into its negative value.
• The y-value remains the same.

In mathematical terms, for a point \((x, y)\), after reflecting about the \(y\)-axis, the new coordinate is \((-x, y)\). When applied to functions like \(y = x^3\), reflecting about the \(y\)-axis translates to replacing \(x\) with \(-x\) in the equation. This principle can be generalized for any function, impacting the entire visual representation of the graph.
Understanding these fundamental transformations helps in analyzing and transforming more complex functions in advanced mathematics.