Question

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

Short answer

\( y = -x^3 \)

Step-by-step solution

Understand the original function

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

Recognize the transformation

To reflect a graph about the \( x \)-axis, we multiply the output (y-value) of the original function by -1.

Apply the reflection

Multiply the function \( y = x^3 \) by -1 to get the reflected function: \( y = -x^3 \).

Key concepts

Cubic Functions

Cubic functions are polynomial functions of degree three. The general form is: \( f(x) = ax^3 + bx^2 + cx + d \). These functions have interesting properties, such as inflection points where the curvature changes direction.
For the simplest cubic function, \( y = x^3 \), there are no additional terms beyond the cubic term. This function has a symmetry point at the origin (0,0) and extends infinitely in all directions.
Some key features of cubic functions include:

• The graph of \( y = x^3 \) is S-shaped, increasing to the right and decreasing to the left.
• It has one inflection point at the origin, where the graph changes concavity.
• The function is odd, meaning that \( f(-x) = -f(x) \).

Understanding these properties helps when applying transformations like reflections.

Graph Transformations

Graph transformations are ways to modify the appearance of a function's graph by altering its equation. These include translations, reflections, stretches, and compressions.
The transformation we are focusing on here is a reflection. Reflections can be about various axes, but in this case, we are dealing with a reflection about the x-axis.
For any function \( y = f(x) \), reflecting it about the x-axis involves multiplying the output by -1, giving \( y = -f(x) \). This flips the graph upside down.
When transforming graphs, remember these key points:

• Reflection about the x-axis changes the sign of the y-values.
• This does not alter the x-values.
• The inflection point of the cubic function \( y = x^3 \) is preserved in its reflected counterpart \( y = -x^3 \).

This specific transformation helps in visualizing the behavior of the function under various conditions.

Reflection About the x-axis

Reflecting a graph about the x-axis means that every point on the graph is flipped over the x-axis. This transformation changes the sign of all y-values, effectively producing a mirror image of the original graph.
For the function \( y = x^3 \), performing this reflection involves:

• Identifying the original function: \( y = x^3 \).
• Multiplying the function by -1: \( y = -x^3 \).

After the reflection, the positive part of the graph above the x-axis moves below the x-axis, and the negative part moves above the x-axis.
This results in the function \( y = -x^3 \), where every y-value is the opposite of the corresponding y-value in \( y = x^3 \). Understanding reflections allows us to predict how graphs will look after this transformation, which is a crucial skill in algebra and calculus.