Quadratic Functions and Graphs

Quadratic Functions and Graphs

Quadratic Functions

  • Quadratic functions are polynomials of the second degree - that means the highest power of the variable (usually represented as ‘x’) that appears in the equation is 2.
  • Standard Form of a quadratic function is f(x) = axe^2 + bx + c. Here, a, b, c are real numbers and a ≠ 0.

Constructing Quadratic Functions

  • To construct a quadratic function, it is necessary to identify three unique points on a plane, then solve the system of equations to find a, b, and c in the standard form.
  • For a quadratic function with roots p and q, the function can also be represented as f(x) = a(x-p)(x-q).

The Graph of a Quadratic Function

  • The graph of any quadratic function is called a parabola.
  • The vertex of a parabola is the point where it turns; hence it represents the maximum or minimum value of the function.
  • If a is positive in the standard form, the parabola opens upwards, and the vertex is the minimum point. If a is negative, then the parabola opens downwards, and the vertex is the maximum point.

Interpreting Quadratic Functions

  • Roots of the quadratic equation axe^2 + bx + c = 0 represent where the graph of the function intersects the x-axis.
  • The axis of symmetry, a vertical line going through the vertex, divides the parabola into two equal halves.

Solving Quadratic Equations

  • Quadratic equations can often be solved by factoring, that is, rewriting the quadratic equation into a product of two binomial expressions set equal to zero.
  • Another way of solving is by completing the square which is used to rewrite the equation in vertex form (a(x-h))^2 + k.
  • Quadratic equations can also be solved using the quadratic formula: x = [-b±sqrt(b^2 - 4ac)]/(2a). The expression under the square root (b² - 4ac) is called the discriminant. Its value can determine the nature of the roots of the quadratic equation.
    • If the discriminant is greater than 0, the equation has two distinct real roots.
    • If the discriminant equals 0, the equation has exactly one real root (a repeated root).
    • If the discriminant is less than 0, the equation has no real roots but two complex roots.

Remember: The procedures for graphing and solving quadratic equations can be applied to any type of quadratic function, whether it is given in standard, vertex, or factored form. Complex roots or solutions refer to results that contain an ‘i’, the square root of -1, in their expressions (a component of complex numbers).