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
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
Different parts of the parabola
Axis of Symmetry/ Vertex
Parabola in Real World
How to write an Equation of A Parabola
Steps:
- 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)
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!