Quadratic Functions

Definition and Basic Properties

Quadratic functions are a fundamental class of functions in algebra, characterized by the presence of an term. Their graphs are known as parabolas, which exhibit reflective symmetry and can open upwards or downwards depending on their coefficients.

• Quadratic Function: Any function of the form , where .

• Parabola: The graph of a quadratic function; it is symmetric about a vertical line called the axis of symmetry.

• Coefficient 'a': Determines the direction of opening:

  • If , the parabola opens upward (minimum point).

  • If , the parabola opens downward (maximum point).

• Vertex: The maximum or minimum point of the parabola; the axis of symmetry passes through the vertex.

Forms of Quadratic Functions

Quadratic functions can be written in two main forms: the general form and the standard form.

• General Form:

• Standard Form:

Finding the Vertex

The vertex is a key feature of the parabola, representing its highest or lowest point. The method to find the vertex depends on the form of the quadratic function.

• General Form:

  • The vertex is found using:

• Standard Form:

  • The vertex is directly from the equation .

Intercepts of Quadratic Functions

Intercepts are points where the graph crosses the axes.

• Y-intercept:

  • Set and solve for .

  • In general form:

  • In standard form: Substitute and simplify.

• X-intercepts:

  • Set and solve for .

  • General form: Use the quadratic formula:

  • Standard form: Use the square root method:

    • Set and solve for .

Graphing Quadratic Functions

Graphing a quadratic function involves several systematic steps to ensure accuracy and clarity.

1. Find the vertex.

2. Determine the direction of opening (up or down).

3. Find the y-intercept.

4. Find the x-intercepts (there may be 2, 1, or 0 real solutions).

5. Plot additional points as needed for shape.

6. Use the reflective property about the axis of symmetry.

Transformations of Quadratic Functions

Quadratic functions can be transformed by shifting, reflecting, stretching, or shrinking their graphs. These transformations are often discussed in detail in function chapters.

• Horizontal Shifts:

  • Left: Replace with ().

  • Right: Replace with ().

• Vertical Shifts:

  • Up: Add to the function ().

  • Down: Subtract from the function ().

• Reflections:

  • Flip over the x-axis by multiplying the function by .

• Stretching/Shrinking:

  • Multiply the function by a constant ; stretches, shrinks.

Examples

• Example 1: Find the vertex of .

  • Vertex:

• Example 2: Find the y-intercept of .

  • Set :

  • Y-intercept:

• Example 3: Find the x-intercepts of .

  • Set and use the quadratic formula:

Additional info: Transformations are covered in detail in Chapter 2.5. The reflective property means that for every point on one side of the axis of symmetry, there is a corresponding point on the other side.