Properties of Quadratic Function

- Indra B. Kshattry PhD

Introduction:

A quadratic function is a type of polynomial function that has a degree of 2. It is generally represented as: $$ f(x) = ax^2 + bx + c $$ Where, a, b, c are constanat numbers and a ≠ 0.
Here are some key properties of quadratic functions, along with examples:

Parabola Shape

If \( a > 0 \), the parabola opens upward; if \( a < 0 \), it opens downward.

Vertex

The vertex is the highest or lowest point of the parabola, depending on the direction it opens.
The vertex is at \( x = \frac{-b}{2a} \). Substitute this into the function to find the y-coordinate, that is, \( y = \frac{4ac - b^2}{4a}\)

Axis of Symmetry

A quadratic function is symmetric about a vertical line called the axis of symmetry.
The axis of symmetry is \( x = \frac{-b}{2a} \).

Roots/Zeros

The roots (or zeros) are the x-values where the parabola intersects the x-axis (f(x) = 0), that is, $$ ax^2 + bx + c = 0 $$ Roots are found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The discriminant (Δ = b² - 4ac) determines the nature of the roots:

• Δ > 0: Two distinct real roots.
• Δ = 0: One real root (the vertex lies on the x-axis).
• Δ < 0: No real roots (the parabola does not cross the x-axis)

Y-Intercept

The y-intercept is the point where the parabola crosses the y-axis (x = 0).
The y-intercept is \( c \) (where \( x = 0 \)).

Domain and Range

The domain is all real numbers. The range depends on the direction of the parabola.

• Domain: The domain of a quadratic function is always all real numbers (-∞, ∞).
• Range: The range depends on the direction of the parabola:
  • If it opens upward (a > 0), the range is [ y-coordinate of vertex, ∞).
  • If it opens downward (a < 0), the range is (-∞, y-coordinate of vertex].