A quadratic function is a function that can be written in the form $ y = ax ^ 2 + bx + c $.
E. g. $ y = 5x^2−4x − 12$ Here a = 5, b = 4 and c = −12
The first term (a) is called quadratic. It is never equal to zero. The second term is linear (b), and the third term is constant (c).
The terms b, c can be 1 or 0. $ y = 4x^2 −16 $ or $ y = x^2 + 10x $ are still quadratic functions.
The graph of a quadratic function is always a parabola. In comparison, a graph of every linear function (y = ax + b) is a straight line. The quadratic term a determines how steeply the function will rise (the higher the absolute value of a is, the steeper the parabola rises). If a is positive, the function is convex (U-shaped - opens upward), if negative, the function is concave (U opens downward).
Linear term b affects the position of the function and c determines at which point the function intersects the y-axis (when we set x=0 we will receive y-coordinate of intersection $ y = a×0^2 + b×0 + c$). The figure below shows functions with different values a, b and c .
To determine at which point the quadratic function intersects the x-axis let y=0 ($ax^2+bx+c=0$). Now it is a quadratic equation which can have two roots (blue, green), one (pink), or none (red).
Each parabola has one vertex. The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the quadratic term a is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape. If the quadratic term a is negative, the vertex will be the highest point on the graph. Determining the coordinates of a vertex can be obtained by converting the function to the vertex/standard form : $ y = a(x − h) + k $ where h, k are the coordinates of the parabola vertex. We do this by completing the square using the formulae:
E.g. for $x^2 + 4x + 1 $ (blue) we will consider $x^2+4x $ for $a^2+2ab $ and we must calculate b from 2ab, then add $b^2$ (and subtract it at the same time), so rewriting the function to the vertex form will look like this:
When $ y = (x + 2)^2−3 $ is compared to $ y = a (x − h) + k $, then the coordinates of the vertex [h, k] can be determined as [−2; −3] .Another option is to use the formula. Vertex coordinates V [h; k] is calculated as: $$
The line of symmetry of quadratic function is the vertical line x = h
Properties of the quadratic function
The domain (set of all x-values) is all real numbers, the range of values (set of all y-values) depends on the parameters of the function, but always goes from the vertex to infinity towards the positive or negative part of the y-axis.
If a < 0, the value range is (− ∞; y-coordinate of the vertex]; if a> 0, it is [y-coordinate of the vertex; ∞).
The quadratic function is even (symmetric about the y-axis) if the linear term (b) is zero.