## Quadratic function

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:

or

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: $$

E.g.

$$**k**(y-coordinate of the vertex) can be also calculated as: $$

E.g.

$$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.