3.1 Quadratic Functions

• Quadratic functions


  Defined by the equation f(x)= ax2 + bx + c 0, where a, b, and c are real numbers and a ≠ 0
• Graphing quadratic functions

  • The graph of y = ax2 + bx + c, a≠0 is a parabola
  • The standard form of the equation is y=a(x-h)²+k
  • The vertex of the parabola is at (h, k)
  • The vertical line passing through the vertex of the parabola is the axis of symmetry of the parabola
  • If a>0, the parabola opens upward and the y-value at the vertex is the minimum value
  • If a<0, the parabola opens downward and the y-value at the vertex is the maximum value
• Finding the vertex of the parabola

  1. x=-b/2a

  2. y=f(x)=a(x-h)²+k
• Maximum and minimum values

  The vertex (h, k) is the maximum or minimum value of the function<|>If a > 0, the parabola opens upward, hence (h, k) is minimum<|>If a <0, the parabola opens downward, hence, (h, k) is maximum
• Graphs of quadratic functions

  • The function y= ax2 is the simplest of a class of quadratic functions
  • The axis of symmetry is the y-axis
  • The graph of a quadratic equation in the form y=a(x-h) is a translation of the graph y=ax²
  • The graph of y= ax²+k is also a translation of the graph y=ax²
  • If a > 0, parabola opens upward; if a < 0, the parabola opens downward
  • The larger the |a|, the narrower the parabola
  • The vertex is (h, k)
  • The axis of symmetry is the vertical line x=h
• Quadratic equation

  An equation that can be written in the form ax2 + bx + c=0, where a ≠ 0
• Roots of a quadratic equation

  The solutions of ax2 + bx+c=0
• Solution set of a quadratic function

  A set of ordered pairs of numbers that satisfy the function<|>If the second number of an ordered pair is zero, then the first number is called a zero of the quadratic function