Parabolic Calculus

Characteristics of a Parabola
A quadratic function, as discussed in the Functions branch, is a function y=f(x) where f(x) = ax2+bx+c.

From conditions (1), (2) and (3) above, we obtain the sketch graph for the domain -2 <= x <= 2.
 * Substituting f(x)=0 will give the x-coordinates where the quadratic function graph intersects the x-axis. Hence the graph have 2 distinct intersection points (1). But the resulting is a quadratic equation, so the discriminant, b2-4ac, determines the position of the graph with respect to the x-axis.
 * Differentiating the function, we get dy/dx = 2ax+b, since the gradient is zero when a curve is at its turning point i.e. dy/dx = 0. We make x the subject of the equation. Then we get the x-coordinate of the turning point,
 * For a line passing through this point and parallel to the y-axis, we get the equation of the axis of symmetry of the graph as x = -(b/2a).
 * Substituting this expression into the equation to arrive at the coordinates of the turning point.
 * D is the quadratic discriminant, b2-4ac. The coordinates of the turning point (whether maximum or minimum) is given by, (2)
 * By completing the square, we express the function as f(x) = a(x-p)2+q. We get
 * The value q defines the maximum or minimum value of the function. Taking [x+(b/2a)]2 >= 0, we solve the resulting inequality for f(x). We get the f(x) as f(x) >= q (minimum value) if a>0 or f(x) <= q (maximum value) if a<0. Hence, the nature of the value of a defines the shape of the graph. (3)

From the diagram, we can conclude that the graph of a quadratic function f(x) = ax2+bx+c is a parabola. Any parabola can be drawn based on the characteristics of the function.