Quadratic Functions

Graphing and Analyzing Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form . Their graphs are called parabolas, which can open upwards or downwards depending on the leading coefficient.

• Standard Form: , where is the vertex of the parabola.

• General Form:

Key Properties of Quadratic Functions

• Direction of Opening: Determined by the sign of in .

  • If , the parabola opens upward.

  • If , the parabola opens downward.

• Vertex: The highest or lowest point on the graph, given by in standard form. In general form, the vertex is at:

• Axis of Symmetry: The vertical line passing through the vertex, .

• Y-intercept: The point where the graph crosses the y-axis, found by evaluating .

• X-intercepts (Roots): The points where . These can be found by factoring, completing the square, or using the quadratic formula:

• Domain: All real numbers, .

• Range: Depends on the direction of opening:

  • If , range is .

  • If , range is .

• Maximum or Minimum Value: The value at the vertex. If the parabola opens upward, $k$ is the minimum; if downward, $k$ is the maximum.

Example: Analyzing

• Direction of Opening: , so the parabola opens downward.

• Vertex:

• Axis of Symmetry:

• Y-intercept:

• X-intercepts: Set :

• Domain:

• Range: Since the parabola opens downward, range is

• Maximum Value: $12x = 3$

Summary Table: Properties of