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: