Quadratic Functions: Finding the Vertex

Introduction

• Continuing with quadratic functions, focusing on finding the vertex of a parabola.
• Parabolas can open up or down.

Standard Form of a Quadratic Function

• Standard form: ( f(x) = a(x - h)^2 + k )
• The vertex of the parabola is at the point ((h, k)).

Example 1

• Function: ( f(x) = -3(x - 2)^2 + 12 )
  • a is negative, indicating the parabola opens down.
  • h = 2, k = 12.
  • Vertex: ((2, 12)).

Example 2

• Function: ( f(x) = -2(x + 4)^2 - 8 )
  • a is negative, parabola opens down.
  • Rewrite (x + 4) as (x - (-4)) to fit the standard form.
  • h = -4, k = -8.
  • Vertex: ((-4, -8)).

Example 3

• Function: ( f(x) = 5(x + 3)^2 + 2 )
  • a is positive, indicating the parabola opens up.
  • Rewrite (x + 3) as (x - (-3)) to fit the standard form.
  • h = -3, k = 2.
  • Vertex: ((-3, 2)).

Key Points

• The sign of a determines the opening direction of the parabola:
  • Negative a: parabola opens down.
  • Positive a: parabola opens up.
• The vertex coordinates are ((h, k)).
• h is derived from (x - h) in the standard form, thus changes sign if it appears as (x + b).
• k is taken as it is from the equation.

Next Steps

• Upcoming topic: Finding the vertex if the equation is not in standard form.