Question

The shape of a
function is a half parabola.

Answer 1

A half parabola can refer to either the upper half or the lower half of a standard parabola. The general form of a parabola is given by the quadratic equation:

\[ y = ax^2 + bx + c \]

Where:
- \( a \) determines the direction and width of the parabola (positive \( a \) means it opens upwards, and negative \( a \) means it opens downwards).
- \( b \) adjusts the position of the vertex along the x-axis.
- \( c \) is the y-intercept.

If you're considering just the upper half of a parabola, you could restrict the function to:

1. For the upward-opening half parabola:
- If you only want the upper half (for example, from the vertex upwards), you might use:
\[
y = a(x - h)^2 + k \quad \text{for} \quad y \geq k
\]
where \( (h, k) \) is the vertex of the parabola.

2. For the downward-opening half parabola:
- If you want just the lower half (from the vertex downwards), you could use:
\[
y = a(x - h)^2 + k \quad \text{for} \quad y \leq k
\]

To illustrate half parabolas:
- Example of an upward-opening half parabola:
\[
y = x^2 \quad \text{for} \quad y \geq 0
\]
This describes the shape of a half parabola starting from the vertex at (0,0) and opening upwards.

- Example of a downward-opening half parabola:
\[
y = -x^2 + 4 \quad \text{for} \quad y \leq 4
\]
This describes a half parabola with its vertex at (0, 4) that opens downwards.

Let me know if you need further details or specific examples!