GH

Chapter 2: Quadratic Functions

  • Lesson 1: Transformations of Quadratic Functions
    • Quadratic Functions
    • A quadratic function is a function that can be written in the form f(x) = a(x - h)2 + k, where a ≠ 0
    • The U-shaped graph of a quadratic function is called a parabola
    • Translations
    • Horizontal Translation
      • f(x) = x2
      • f(x - h) = (x - h)2
      • Shifts left when h < 0
      • Shifts right when h > 0
    • Vertical Translation
      • f(x) = x2
      • f(x) + k = x2 + k
      • Shifts down when k < 0
      • Shifts up when k > 0
    • Reflections
    • Reflection in the x-axis
      • f(x) = x2
      • -f(x) = -(x2) = -x2
    • Reflection in the y-axis
      • f(x) = x2
      • f(-x) = (-x)2 = x2
    • Stretches and Shrinks
    • Horizontal Stretches and Shrinks
      • f(x) = x2
      • f(ax) = (ax)2
      • Horizontal stretch (away from y-axis) when 0 < a < 1
      • Horizontal shrink (towards y-axis) when a > 1
    • Vertical Stretches and Shrinks
      • f(x) = x2
      • a * f(x) = ax2
      • Vertical stretch (away from x-axis) when a > 1
      • Vertical shrink (towards x-axis) when 0 < a < 1
    • Vertex Form
    • The lowest part on a parabola that opens up or the highest point on a parabola that opens down is called the vertex
    • The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where a ≠ 0 and the vertex is (h, k)
  • Lesson 2: Characteristics of Quadratic Functions
    • Quadratic Function Forms
    • Axis of Symmetry - a line that divides a parabola into mirror images and passes through the vertex; the axis of symmetry is the vertical line x = h or x = -b/2a
    • Standard Form - a quadratic function written in the form f(x) = ax2 + bx + c
    • Intercept Form - a quadratic function in the form f(x) = a(x - p)(x - q)
    • Properties of the Standard Form
    • The parabola opens up when a > 0 and opens down when a < 0
    • The graph is narrower than the graph of f(x) = x2 when |a| > 1 and wider when |a| < 1
    • The axis of symmetry is x = -b/2a and the vertex is (-b/2a, f(-b/2a))
    • The y-intercept is c; the point (0, c) is on the parabola
    • Minimum and Maximum Values
    • For the quadratic function f(x) = ax2 + bx + c, the y-coordinate of the vertex is the minimum value of the function when a > 0 and the maximum value when a < 0
    • a > 0
      • Minimum value: f(-b/2a)
      • Domain: all real numbers
      • Range: y ≥ f(-b/2a)
      • Decreasing to the left of x = -b/2a
      • Increasing to the right of x = -b/2a
    • a < 0
      • Maximum value: f(-b/2a)
      • Domain: all real numbers
      • Range: y ≤ f(-b/2a)
      • Increasing to the left of x = -b/2a
      • Decreasing to the right of x = -b/2a
    • Properties of the Intercept Form
    • Because f(p) = 0 and f(q) = 0, p and q are the x-intercepts of the graph of the function
    • The axis of symmetry is halfway between (p, 0) and (q, 0); x = (p + q)/2
    • The parabola opens up when a > 0 and opens down when a < 0
  • Lesson 3: Modeling with Quadratic Equations
    • Writing Quadratic Equations
    • Given a point and the vertex (h, k), use vertex form: y = a(x - h)2 + k
    • Given a point and x-intercepts p and q, use intercept form y = a(x - p)(x - q)
    • Given three points, write and solve a system of three equations in three variables