y = (x - 1)^3

The expression \( y = (x - 1)^3 \) represents a cubic function. Here are some key points and properties about this function:

1. Basic Properties:

• Cubic Function: The function \( y = (x - 1)^3 \) is a cubic polynomial, which means it has the general form \( y = ax^3 + bx^2 + cx + d \). In this case, it is centered around \( x = 1 \).
• Shape: The graph of this function has an inflection point at \( x = 1 \) and will have a characteristic "S" shape.

2. Roots:

• Finding Roots: To find the roots of the function, set \( y = 0 \): \[ (x - 1)^3 = 0 \implies x - 1 = 0 \implies x = 1 \]
• Thus, the root (or zero) of the function is at \( x = 1 \).

3. Derivative:

• To find critical points and analyze the function further, we can take the derivative: \[ y' = 3(x - 1)^2 \]
• This derivative tells us that \( y' = 0 \) when \( x = 1 \). At this point, there's a local extremum.

4. Second Derivative:

• The second derivative helps us determine the concavity: \[ y'' = 6(x - 1) \]
• • If \( x < 1 \), then \( y'' < 0 \) (the function is concave down).
• • If \( x > 1 \), then \( y'' > 0 \) (the function is concave up).
• • At \( x = 1 \), the second derivative is zero, indicating a point of inflection.

5. Graphical Representation:

• The graph of this function will pass through the point (1, 0) and will exhibit the behavior characteristic of cubic functions, increasing indefinitely as \( x \) approaches positive and negative infinity but flattening out near the root.

6. Behavior at Infinity:

• As \( x \to \infty \), \( y \to \infty \).
• As \( x \to -\infty \), \( y \to -\infty \).

Summary:

The function \( y = (x - 1)^3 \) characterizes a cubic polynomial with a root at \( x = 1 \), an inflection point at the same value, and displays typical cubic behavior in terms of growth and shape. Understanding these key features will help in analyzing and graphing the function effectively.