Quadratic and Polynomial Functions: Factoring, Quadratic Formula, and Graphing

A quadratic function is a polynomial function of degree 2. Its graph is a parabola. The standard form is y = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex. The factored form is y = a(x - r_1)(x - r_2), where r_1 and r_2 are the x-intercepts (zeros or roots).

Factoring a quadratic expression means writing it as a product of two linear expressions. Common methods include common factoring (e.g., ax^2 + bx = x(ax + b)), factoring simple trinomials (x^2 + bx + c), factoring complex trinomials (ax^2 + bx + c, often using decomposition), and factoring the difference of squares (a^2 - b^2 = (a - b)(a + b)). Factoring helps find the zeros of the function.

The quadratic formula is used to find the roots (or zeros) of any quadratic equation in the form ax^2 + bx + c = 0. It can be used when factoring is difficult or impossible. The solutions are given by the formula.

The discriminant is the part of the quadratic formula under the square root sign, Δ = b^2 - 4ac. It determines the nature of the roots of a quadratic equation: If Δ > 0, there are two distinct real roots. If Δ = 0, there is one real root (or two equal real roots). If Δ < 0, there are no real roots (two complex conjugate roots).

To graph a quadratic function, identify key features: the direction of opening (up if a > 0, down if a < 0), the vertex (the turning point of the parabola, (h, k) from vertex form or x = -b/(2a) for the x-coordinate), the axis of symmetry (the vertical line x = h or x = -b/(2a)), the y-intercept (set x = 0, which is 'c' in standard form), and the x-intercepts (set y = 0, solve by factoring or quadratic formula).

A polynomial function is a function of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n ≠ 0, n is a non-negative integer, and a_i are real coefficients. The highest exponent, n, is the degree of the polynomial. The coefficient a_n is the leading coefficient. The degree and leading coefficient determine the end behaviour of the graph (how the graph behaves as x approaches positive or negative infinity).