Parabola

A parabola is the graph associated with a quadratic function, i.e. a function of the form \(y=ax^2+bx+c\).




The general or standard form of a quadratic function is \(y =ax^2+bx+c\), or in function form, \(f(x)=ax^2+bx+c\), where \(x\) is the independent variable, \(y\) is the dependent variable, and \(a\), \(b\), and \(c\) are constants.

• The larger the absolute value of \(a\), the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
  
• If \(a\) is positive, the parabola opens upward, if negative, the parabola opens downward.

x-intercepts: The x-intercepts, if any, are also called the roots of the function. The x-intercepts are the solutions to the equation \(0=ax^2+bx+c\) and can be calculated by the formula:
\(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\)

Expression \(b^2-4ac\) is called discriminant:

• If discriminant is positive parabola has two intercepts with x-axis;
• If discriminant is negative parabola has no intercepts with x-axis;
• If discriminant is zero parabola has one intercept with x-axis (tangent point).

y-intercept: Given \(y =ax^2+bx+c\), the y-intercept is \(c\), as y intercept means the value of y when x=0.

Vertex: The vertex represents the maximum (or minimum) value of the function, and is very important in calculus.

The vertex of the parabola is located at point \((-\frac{b}{2a},\) \(c-\frac{b^2}{4a})\).
Note: typically just \(-\frac{b}{2a},\) is calculated and plugged in for x to find y.