Recall that a quadratic function has the form

$\displaystyle f(x)=ax^{2}+bx+c$. 

where $a$, $b$, and $c$ are constants, and $a\neq 0$.

The graph of a quadratic function is a U-shaped curve called a parabola.  This shape is shown below.

Parabola 

The graph of a quadratic function is a parabola.

In graphs of quadratic functions, the sign on the coefficient $a$ affects whether the graph opens up or down. If $a<0$, the graph makes a frown (opens down) and if $a>0$ then the graph makes a smile (opens up). This is shown below.

Direction of Parabolas

The sign on the coefficient $a$ determines the direction of the parabola. 

Features of Parabolas

Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.

Vertex

One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. 

Axis of Symmetry

Parabolas also have an axis of symmetry, which is parallel to the y-axis. The axis of symmetry is a vertical line drawn through the vertex.

$y$-intercept

The y-intercept is the point at which the parabola crosses the y-axis. There cannot be more than one such point, for the graph of a quadratic function. If there were, the curve would not be a function, as there would be two $y$ values for one $x$ value, at zero.

$x$-intercepts

The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of $x$ at which $y=0$. There may be zero, one, or two $x$-intercepts. The number of $x$-intercepts varies depending upon the location of the graph (see the diagram below).

Possible $x$-intercepts

A parabola can have no x-intercepts, one x-intercept, or two x-intercepts.

Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the function. These are the same roots that are observable as the $x$-intercepts of the parabola.

Notice that, for parabolas with two $x$-intercepts, the vertex always falls between the roots. Due to the fact that parabolas are symmetric, the $x$-coordinate of the vertex is exactly in the middle of the $x$-coordinates of the two roots.