Algebra 1: Quadratic Functions and Parabolas

Quadratic Functions and Parabolas Study Notes

• The axis of symmetry for a quadratic function can be calculated using the formula: ( x = -\frac{b}{2a} ).

• The equation to solve and find the roots of a quadratic equation is given by the quadratic formula: ( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

• To find the vertex of a quadratic function represented as ( ax^2 + bx + c ), substitute the ( x ) value back into the equation to get the corresponding ( y ) value.

• The axis of symmetry ensures that both sides of the parabola are mirror images; if this is not true, it indicates an error in calculations.

• The vertex of a parabola is the minimum or maximum point located on the axis of symmetry, essential for determining the parabola's direction.

• The direction in which a parabola opens (upward or downward) depends on the coefficient of ( x^2 ):

  • If positive, the parabola opens upwards.
  • If negative, it opens downwards.

• The roots of a quadratic function represent the points where the graph intersects the x-axis, also known as the x-intercepts.

• The discriminant of a quadratic equation determines the nature of the roots:

  • Two distinct real roots when positive,
  • One real root (double root) when zero,
  • No real roots when negative.

• In the formula ( h = -16t^2 + vt + s ):

  • ( H ) represents the initial height,
  • ( T ) indicates time,
  • ( V ) is the initial velocity,
  • ( S ) stands for speed.