Introduction to Quadratic Functions

Key Characteristics of Quadratic Functions

• Parabola: The U-shaped graph of a quadratic function. Can be right-side-up or upside-down.
• Symmetry: Quadratic functions are symmetric. The line that divides the parabola into two congruent halves is called the axis of symmetry.
• Vertex: The highest or lowest point on the parabola.
• X-Intercepts/Roots/Zeros/Solutions: Points where the graph crosses the x-axis.
  • X-intercept occurs when y = 0.

Forms of Quadratic Functions

• Standard Form: ( ax^2 + bx + c )
  • a: Coefficient of ( x^2 )
  • b: Coefficient of ( x )
  • c: Constant term
• Vertex Form: ( a(x - h)^2 + k )
  • Vertex is at ((h, k))

Linear vs Quadratic Parent Functions

• Linear Function: ( f(x) = x )
  • Passes through origin with a slope of 1.
• Quadratic Parent Function: Takes every input value and squares it.
  • Graph is always above the x-axis because squared values are non-negative.

Axis of Symmetry

• Definition: Vertical line dividing the parabola into two congruent halves.
• Formula: ( x = \frac{-b}{2a} )
• Example: Graph a quadratic with ( x = 4 ) as the axis of symmetry.

Vertex

• Definition: Highest or lowest point of the graph.
• Finding Vertex:
  • X-value is ( \frac{-b}{2a} ).
  • Y-value: Plug x-value into the function.

Determining Parabola Orientation

• Parabola Opens Up:
  • ( a > 0 )
  • Vertex is a minimum.
• Parabola Opens Down:
  • ( a < 0 )
  • Vertex is a maximum.

Examples

• Example 1: ( f(x) = x^2 + 10x + 15 )

  • Identify ( a = 1 ), ( b = 10 ), ( c = 15 ).
  • Axis of symmetry: ( x = -5 ).
  • Vertex: ((-5, -10)).
  • Parabola opens up.

• Example 2: ( f(x) = -2x^2 - 8x - 15 )

  • Identify ( a = -2 ), ( b = -8 ), ( c = -15 ).
  • Axis of symmetry: ( x = -2 ).
  • Vertex: ((-2, -7)).
  • Parabola opens down.

• Example 3: No B Value, ( f(x) = x^2 - 5 )

  • Identify ( a = 1 ), ( b = 0 ), ( c = -5 ).
  • Axis of symmetry: ( x = 0 ).
  • Vertex: ((0, -5)).

This concludes the notes on the introduction to quadratic functions. This summary should help in understanding the basic concepts and how to approach quadratic equations.