Lecture Notes: Introduction to Parabolas

What is a Parabola?

• A common type of curve in mathematics.
• The term "parabola" has Greek origins:
  • "Parah" = beside/alongside.
  • "Bola" = related to ballistics (throwing).
• Often associated with the trajectory of thrown objects in physics.

Properties of Parabolas

• Parabolas can open either upwards or downwards:
  • Open Upwards: Resembles a right-side-up U shape.
  • Open Downwards: Resembles an upside-down U shape.

Vertex

• The vertex is the maximum or minimum point of the parabola:
  • For upward-opening parabolas, it is the minimum point.
  • For downward-opening parabolas, it is the maximum point.
  • Example:
    • Yellow parabola vertex: (3, -3.5)
    • It has no maximum as it opens upwards.

Axis of Symmetry

• A line that divides the parabola into two symmetrical halves.
• It passes through the vertex:
  • For yellow parabola: x = 3.5
  • For pink parabola: x = -1
  • For green parabola: x = -6

Intercepts

• Y-Intercept: Where the curve intersects the Y-axis.
  • Yellow parabola Y-intercept: (0, 3)
  • Pink parabola Y-intercept is not visible but exists outside the visible area.
• X-Intercepts: Where the curve intersects the X-axis.
  • Yellow parabola X-intercepts: (1, 0) and (6, 0)
  • Pink parabola has no X-intercepts because it's above the X-axis.
  • X-intercepts are symmetric around the axis of symmetry.

Key Observations

• A parabola can intersect the X-axis in:
  • Zero points (above the axis).
  • One point (touching the axis).
  • Two points (crossing the axis).
• The symmetry of X-intercepts is a notable feature.

Future Topics

• In subsequent lectures, we will dive deeper into algebraic representations of parabolas:
  • Most basic form: y = x²
  • More complex forms: e.g. y = 2x² - 5x + 7
• Equations involving second-degree terms represent parabolas, often referred to as quadratics.