Problem 60 - Elementary Algebra

Question

\(y=-x^{3}\)

Step-by-Step Solution

Answer

The function \( y = -x^3 \) is a decreasing cubic curve inverted around the y-axis.

Step 1: Understand the Equation

The given equation is a cubic function represented by \( y = -x^3 \). This tells us that for any input \( x \), \( y \) is the result of the negative of \( x \) raised to the power of 3.

Step 2: Identify Key Characteristics

The cube function \( y = x^3 \) normally increases as \( x \) becomes more positive and decreases as \( x \) becomes more negative. However, because our function is \( y = -x^3 \), it is reflected over the x-axis, so it decreases as \( x \) goes from negative to positive.

Step 3: Graph Analysis

To understand the behavior, imagine the graph of \( y = -x^3 \). It passes through the origin (0,0) because when \( x = 0 \), \( y = 0 \). For \( x > 0 \), \( y < 0 \), indicating a downward curve. For \( x < 0 \), \( y > 0 \) showing an upward curve.

Step 4: Determine Key Points

To graph or understand the behavior properly, determine a few key points: - When \( x = 0 \), \( y = 0 \) - When \( x = 1 \), \( y = -1 \) - When \( x = -1 \), \( y = 1 \) - When \( x = 2 \), \( y = -8 \) - When \( x = -2 \), \( y = 8 \)

Step 5: Conclusion on Behavior

The function \( y = -x^3 \) is continuously decreasing as \( x \) increases, emphasizing its cubic nature with inversion along the y-axis. This determines its overall shape, steepness, and directionality as a typical cubic curve, but inverted.

Key Concepts

Graph Analysis

Graph analysis is about understanding the behavior and shape of a function by observing its graph. For the cubic function given by \( y = -x^3 \), we observe some unique characteristics. Typically, the standard cubic function \( y = x^3 \) creates an S-shaped curve. It passes through the origin (0,0) and symmetrically curves upwards in the right half-plane and downwards in the left half-plane.

In comparison, the graph of \( y = -x^3 \) is a reflection of this shape over the x-axis. This means the curve forms a mirrored image. The downward curve occurs when \( x > 0 \), while the upward curve is for \( x < 0 \).

• This function intersects the origin (0,0), where both \( x \) and \( y \) equal zero.
• For positive \( x \), the value of \( y \) becomes negative, indicating a downward trajectory.
• For negative \( x \), \( y \) becomes positive, showing an upward trajectory.

The graph thus tells us that as \( x \) increases, \( y \) decreases, reflecting the inverse cubic nature of the function.

Reflected Graph

Reflection in graphs is a transformation that flips the graph over a specified axis. In the equation \( y = -x^3 \), the negative sign in front of \( x^3 \) plays a crucial role. It indicates that every point on the standard cubic function \( y = x^3 \) has been reflected over the x-axis.

This transformation means:

• If a point was originally in the first quadrant (positive x, positive y), reflecting it results in the fourth quadrant (positive x, negative y).
• Similarly, points in the third quadrant (negative x, negative y) would move to the second quadrant (negative x, positive y).

Understanding reflections helps students see how transformations affect a graph's position, symmetry, and shape, especially important for math problems involving transformations.

Negative Exponentiation

Negative exponentiation involves raising a number to a power that turns the result negative, directed by the negative sign preceding \( x^3 \) in the function \( y = -x^3 \). This does not affect the position of \( x \) itself but directs \( y \) to its negative equivalent.

For a clearer understanding, consider a few specific calculations:

• For \( x = 1 \), \( y = -(1^3) = -1 \).
• For \( x = -1 \), \( y = -(-1^3) = 1 \).
• For \( x = 2 \), \( y = -(2^3) = -8 \).
• For \( x = -2 \), \( y = -(-2^3) = 8 \).

This reflects a pattern where the exponential operation has initially calculated \( x^3 \), and then the negative sign inversely transforms each \( y \) value. Students often find this important in noticing how subtle changes in equations can lead to significant differences in graphical behavior.