[math]y = a(x-h)^2 +k[/math] is the vertex form of a quadratic equation. The vertex is the point [math](h, k)[/math] on the quadratic. The vertex is the minimum or maximum of the quadratic. [math]y = ax^2 + bx +c[/math] is the standard form of a quadratic equation. Move slider [math]a[/math]. See the width of the quadratic change. See the direction of the quadratic change when [math]a[/math] is a negative number. Positive [math]a[/math] give the quadratic a minimum point. Negative [math]a[/math] give the quadratic a maximum point. Move slider [math]h[/math]. See the horizontal position of the quadratic change. Positive [math]h[/math] values keep the minimum or maximum of the quadratic to the right of the [math]y[/math]-axis. Negative [math]h[/math] values keep the minimum or maximum of the quadratic to the left of the [math]y[/math]-axis. Move slider [math]k[/math]. See the vertical position of the quadratic change. Positive [math]k[/math] values keep the minimum or maximum of the quadratic above the [math]x[/math]-axis. Negative [math]k[/math] values keep the minimum or maximum of the quadratic above the [math]x[/math]-axis.

The real roots of a quadratic are the [math]x[/math] values of the points where the quadratic intersects the [math]x[/math]-axis.

[math]y=ax^2 +bx +c[/math] is the standard form of a quadratic equation. Move quadratic [math]b[/math]. Quadratic [math]b[/math] has a maximum point since its [math]ax^2[/math] term in negative. Move quadratic [math]c[/math]. Quadratic [math]c[/math] has a minimum point since its [math]ax^2[/math] term in positive. The dashed line going through the maximum or minimum point is the quadratic's axis of symmetry. The axis of symmetry divided the quadratic into equal halves.

A quadratic equation can represent the trajectory of an object thrown in the air. A trajectory is the path of an object while it is in the air, ending when it hits the ground or intended target. Point A gives the starting height of the object the millisecond it is released. The [math]y[/math] value of point A is the starting height. Point B is the vertex of the quadratic. Point B gives the maximum height of the object in the air. The [math]y[/math] value of point B is the maximum height. Point C is one of the roots of the quadratic. Point C gives the maximum horizontal distance of the object. The [math]x[/math] value of point C is the total distance the object was thrown. When point C is on the [math]x[/math]-axis, it is considered ground level. Move sliders [math]a[/math], [math]b[/math] or [math]c[/math]. Slider [math]a[/math] is the coefficient in the term [math]ax^2[/math]. Slider [math]b[/math] is the coefficient in the term [math]bx[/math]. Slider [math]c[/math] is the constant term (also the coefficient in the term [math]cx^0[/math]).