Parabolas

Important parts of the parabola

y = ax^2 + bx + c

  • a > 0 = opens upward --> has a minimum
  • a<0 = opens downward ---> has a maximum

  • Absolute value of a > 1 --> narrower parabola
  • Absolute value of a<1 --> wider parabola


  • c > 0--> moves the parabola up the y axis
  • c <0 --> moves the parabola down the y axis


Vertex/Axis of Symmetry

Vertex: Maximum or Minimum point of a parabola
Axis of Symmetry: Vertical line going through the vertex (x value of the vertex point)

To find the axis of symmetry (x value of vertex)
1. Quadratic needs to be foiled out into standard form
2. x = -b/2a
3. Once you have the x value, plug into the equation to calculate the y value of the vertex


Parabola Parts

How to write an Equation of A Parabola

Steps:

  1. Given x intercepts of (2, 0) and (-3,0)

  • y = (x - 2)(x + 3)


2. Given direction graph opens:

  • Upward = a
  • Downward = -a
  • y = a(x - 2)(x + 3)


3. Given a min/max point (vertex!):

  • We are looking for 'k' (it will go in front of factored quadratic)

  • Write out intercepts like step 1: y = (x -2) (x + 3)
  • Given the maximum or minimum point (plug in x and y and solve for k)
Ex: Max of (-0.5, 12.5)
12.5 = k [ (-0.5 - 2) (-0.5 + 3) ]

12.5 = k [ (-2.5)(2.5)]

12.5 = k [-6.25]

12.5 = -6.25k

k = -2

  • So: a parabola with x-intercepts of 2 and -3 with a maximum of (-0.5,12.5) will have an equation of: y = -2(x - 2)(x + 3)


Test it on your calculator!