📐 Function y = x³. Power Function with Odd Exponent
Степенная функция с нечётным показателем
1. Review: What Are Power Functions?
A power function (степенная функция) has the general form:
y = xⁿ, where n is a natural number
✅ Already Studied: Even Exponent
y = x² (parabola)
- Symmetric about the y-axis
- Always non-negative (y ≥ 0)
- U-shaped curve
🆕 Now Studying: Odd Exponent
y = x³ (cubic parabola)
- Symmetric about the origin
- Takes both positive and negative values
- S-shaped curve
📋 Three Key Properties of y = x³
Property 1: Value at Zero
When the argument equals zero, the function value is also zero:
If x = 0, then y = 0³ = 0
This means the graph of y = x³ passes through the origin (0, 0).
Property 2: Sign of the Function
The function preserves the sign of the argument:
x > 0 → y = x³ > 0
Positive input → Positive output
Positive input → Positive output
x < 0 → y = x³ < 0
Negative input → Negative output
Negative input → Negative output
Example: x = 2 → y = 8 (positive); x = −3 → y = −27 (negative)
Property 3: Opposite Arguments → Opposite Values
For any value a:
If x = a, then y = a³
If x = −a, then y = (−a)³ = −a³
If x = −a, then y = (−a)³ = −a³
Opposite inputs always give opposite outputs!
| x | y = x³ | −x | y = (−x)³ | Relationship |
|---|---|---|---|---|
| 2 | 8 | −2 | −8 | Opposite ✓ |
| 3 | 27 | −3 | −27 | Opposite ✓ |
| 0.5 | 0.125 | −0.5 | −0.125 | Opposite ✓ |