The quadratic function given is [tex]\( F(x) = x^2 \)[/tex].
To describe this function, let's break down its characteristics step-by-step:
1. Form of the Function: [tex]\( F(x) = x^2 \)[/tex] is in the standard form of a quadratic function, which is [tex]\( y = ax^2 + bx + c \)[/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. In this case, [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Vertex: For the function [tex]\( F(x) = x^2 \)[/tex], the vertex is the point where the function reaches its minimum value (or maximum value if [tex]\( a < 0 \)[/tex]). Since [tex]\( b = 0 \)[/tex] and [tex]\( c = 0 \)[/tex], the vertex is at the origin [tex]\((0, 0)\)[/tex].
3. Direction: Since [tex]\( a = 1 \)[/tex] which is positive, the parabola opens upwards. If [tex]\( a \)[/tex] were negative, the parabola would open downwards.
4. Parent Function: The function [tex]\( F(x) = x^2 \)[/tex] is considered a "parent function" for quadratic functions. This means it is the simplest form of a quadratic equation.
5. Quadrants:
- The vertex lies at the origin, which technically belongs to all quadrants as it is the point [tex]\((0,0)\)[/tex].
- As the function [tex]\( F(x) = x^2 \)[/tex] opens upwards and increases both to the left and right of the vertex, it predominantly occupies Quadrants I and II.
Given these characteristics, the best description of the function [tex]\( F(x) = x^2 \)[/tex] from the options provided is:
OC. A parent function.
To summarize:
- It is not an inverse function.
- It does not open to the right.
- It is a parent function.
- It opens upwards (not downwards).
- The vertex is at the origin.
- It primarily occupies Quadrants I and II.
