Characteristics of Parabolas

Learning Outcomes

  • Characteristics of Parabolas

    • Identify key features such as:

      • Vertex

      • Axis of symmetry

      • y-intercept

      • Minimum or maximum value of a parabola from its graph.

    • Recognize a quadratic function in both general and vertex form.

    • Calculate the vertex from a quadratic function in general form.

    • Define the domain and range by determining if the vertex is a maximum or minimum point on the graph.

Characteristics of Parabolas

  • A parabola is a U-shaped curve that represents the graph of a quadratic function.

  • The vertex is a critical feature of the graph:

    • If it opens upwards, the vertex is the minimum point.

    • If it opens downwards, the vertex is the maximum point.

  • The vertex acts as a turning point and the graph has a line of symmetry through it, called the axis of symmetry.

Graphical Features

Intercepts

  • y-intercept: where the parabola crosses the y-axis.

  • x-intercepts: where the parabola crosses the x-axis, also known as zeros or roots of the function.

Tips for Success

  • The points where the graph intersects the horizontal axis indicate that the function value equals zero. These can be called horizontal intercepts or zeros.

Equations of Quadratic Functions

General Form

  • A quadratic function can be expressed as:

    • f(x) = ax² + bx + c

  • Where a, b, and c are real numbers with a ≠ 0.

  • Direction of the parabola:

    • If a > 0, the parabola opens upwards.

    • If a < 0, the parabola opens downwards.

Axis of Symmetry

  • Calculated using the formula:

    • x = -b / (2a)

Finding x-intercepts and Y-intercept

  • To find the x-intercepts (roots), solve the equation:

    • ax² + bx + c = 0

  • Midpoint for x-intercepts:

    • The x-coordinate of the vertex is the average of the x-intercepts.

Vertex Form of Quadratic Functions

Standard Form

  • Another form of expressing a quadratic function is:

    • f(x) = a(x - h)² + k

  • Where (h, k) is the vertex of the parabola.

Domain and Range of Quadratic Functions

  • Domain: All real numbers.

  • Range:

    • If a > 0 (parabola opens upwards): y ≥ k

    • If a < 0 (parabola opens downwards): y ≤ k

Examples

  1. To find the vertex of a function, rearrange to vertex form or use the coordinates determined by the formulas mentioned.

  2. Always determine the maximum or minimum value based on the parabola's orientation.