Quadratic functions

A quadratic function is a type of function described by an equation in the form f(x) = ax² + bx + c, where a is not equal to 0. The condition that ‘a’ must not be 0 is crucial, because if a were 0, the equation would reduce to f(x) = bx + c, which is a linear function. The name “quadratic” is related to the term “square”.

The graph of any quadratic function is always a parabola.

Important observations and features of quadratic functions and their graphs include:

• The coefficient ‘a’:
  • If a is greater than 0 (a > 0), the function opens upwards and has a minimum value.
  • If a is less than 0 (a < 0), the function opens downwards and has a maximum value.
• Axis of Symmetry: All parabolas have an axis of symmetry. If the graph paper containing the parabola is folded along this line, the portions on either side will match exactly. The equation of the axis of symmetry is x = -b/(2a). This can be shown through the method of completing the square.
• Vertex: The point where the axis of symmetry intersects the parabola is called the vertex.
  • The x-coordinate of the vertex is -b/(2a).
  • The y-coordinate of the vertex is f(-b/(2a)).
  • The y-coordinate of the vertex represents the minimum or maximum value attained by the function. When the slope of the function is zero, it corresponds to the vertex where the function assumes its minimum or maximum value.
• Y-intercept: The point where the graph crosses the y-axis is called the y-intercept. This value is given by c (when x = 0).
• Graphing: A quadratic function can be graphed by plotting ordered pairs (x, f(x)) that satisfy the function. A recommended approach is to find the axis of symmetry, the y-intercept, and then use the axis of symmetry to find another point, and join these points with a smooth curve.
• Domain and Range:
  • The domain of a quadratic function f(x) = ax² + bx + c is the entire real line. We can take the square of any real number as input.
  • The range of a quadratic function is a subset of the codomain (the set of possible output values). The range depends on whether the parabola opens up or down and the y-coordinate of the vertex. If a > 0, the range is from the minimum value (y-coordinate of the vertex) upwards to positive infinity. If a < 0, the range is from negative infinity up to the maximum value (y-coordinate of the vertex).
• Slope: Unlike linear functions which have a constant slope, the slope of a quadratic function is variable. For a function f(x) = ax² + bx + c, the slope at any point x is given by 2ax + b. Setting the slope equal to 0 (2ax + b = 0) gives x = -b/(2a), which is the x-coordinate of the vertex.

Quadratic functions are closely related to quadratic equations. A quadratic equation is formed when a quadratic function is set equal to a value. The standard form of a quadratic equation is ax² + bx + c = 0, where a is not equal to 0, and a, b, c are integers.

The solutions to a quadratic equation are called roots of the equation. These roots are also known as the zeros of the related quadratic function. The zeros of a quadratic function are its x-intercepts.

Methods for solving quadratic equations include:

1. Graphing: Plotting the associated quadratic function and finding where it intersects the x-axis (the x-intercepts). This method is good for verifying results, but less precise if solutions are not integers.
2. Factoring: Writing the quadratic polynomial as a product of binomials and using the Zero Product Property to set each factor equal to zero and solve for x. The intercept form of a quadratic function, y = a(x-p)(x-q), directly shows the x-intercepts (p and q).
3. Completing the Square: Transforming the equation to have a perfect square on one side and a constant on the other, then taking the square root of both sides.
4. Quadratic Formula: A general formula derived from the method of completing the square. The roots of ax² + bx + c = 0 are given by x = (-b ± √(b² - 4ac)) / (2a). The term b² - 4ac is called the discriminant. The discriminant indicates the number and type of real roots:
   • If b² - 4ac > 0, there are two real roots.
   • If b² - 4ac = 0, there is one real (repeated) root.
   • If b² - 4ac < 0, there are no real roots.