What is the quadratic parent function?

The quadratic parent function is a basic quadratic function that serves as a template for all other quadratic functions. It is represented by the equation f(x) = x^2.

The graph of the quadratic parent function is a U-shaped curve, known as a parabola. It is symmetric around the y-axis and has a single vertex at the origin (0, 0). The vertex is the lowest point on the curve, also known as the minimum point.

The graph of the quadratic parent function extends infinitely in both the positive and negative x-directions. As x approaches positive or negative infinity, f(x) also approaches infinity.

The quadratic parent function is characterized by several key features:

1. Vertex: The vertex of the quadratic parent function is at (0, 0), which represents the minimum point of the parabola.

2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola. For the quadratic parent function, the axis of symmetry is the y-axis (x = 0).

3. Discriminant: The discriminant is the part of the quadratic formula that determines the nature of the solutions to a quadratic equation. For the quadratic parent function, the discriminant is equal to zero, indicating that it has only one real and repeated root.

4. Roots: The quadratic parent function (x^2) has a single root at x = 0. This means that the parabola intersects the x-axis only at the point (0, 0).

5. Graph: The graph of the quadratic parent function is a smooth, symmetric, and continuous curve. It opens upwards since the coefficient of x^2 is positive.

The quadratic parent function is widely used as a reference when studying quadratic equations and functions. By manipulating the coefficients and constants of the parent function, different shapes, positions, and characteristics of quadratic functions can be explored and analyzed.