## Summary and examples

The general equation of a quadratic function is:

, and are called parameters. , because otherwise it is no quadratic function.

The graph of a quadratic function is a parabola. There are two types, one that opens up and one that opens down, depending on the parameter .

: opens up

: opens down

A parabola intersects the -axis in the point . You can easily see this, because for any point on the -axis we have . When is substituted in the equation we get .

When a parabola intersects the -axis we can calculate the intersection points in various ways. The -formula, explained in Quadratic equations (abc-formula), always gives a solution. Another way is based on factorizing the equation, see the topic Quadratic equations (factorizing). This is faster, though not always possible.

The intersection points can be calculated by applying the -formula:

In this formula the discrriminant plays an important role:

We have:

- : the parabola has two different intersection points with the -axis;
- : the parabola has two coinciding intersection points with the -axis (usually it is said that the parabola has just one intersection point); and
- : the parabola has no intersection points with the -axis.

A parabola is a symmetric graph and has a symmetry axis, namely the line:

Then it is clear that the top of the parabola is on this symmetry axis:

This can also be deduced from the -formula. This optimum can also be found by differentiating the function and make the derivative equal to :

see also the topic Differentiation of standard functions .

#### Example 1

Given the function:

1. Determine the intersection point of the graph with the -axis;

2. Does this graph have an intersection point with the -axis, and if so, how many and determine their coordinates.

3. Determine the symmetry axis.

1. Because in the formula is equal to , the intersection point with the -axis is equal to .

2. We have to compute the discriminant:

is greater than en thus the graph has two intersection points with the -axis. The -coordinates of these intersection points are:

so:

and the coordinates are:

and

3. The symmetry axis can be calculated in two way. First by using the formula above, but also by taking the mean value of the two intersection points, namely:

#### Example 2

Given the function:

1. Determine the intersection point of the graph with the -axis;

2. Does this graph have an intersection point with the -axis, and if so, how many and determine their coordinates.

3. Determine the symmetry axis.

1. Because in the formula is equal to , the intersection point with the -axis is equal to .

2. We can compute the discriminant to find out whether there are one or more intersection points, but another way is faster. The equation can be factorized:

and thus we may conclude that the graph has two intersection points with the -axis: en .

3. The symmetry axis lies just in the middle of these two points, so the symmetriy-axis is .

#### Example 3

For which value of does the parabola:

have two coinciding intersection points with the -axis.

These intersection points coincide if the discriminant , so: