|2.3 Quadratic Functions|
So far, we have seen how to graph linear functions, i.e. functions whose graphs are straight lines. In this section, we will discuss a type of function that is in a sense the next step up. Straight lines are created by functions whose highest power of x is 1, and here we will discuss functions whose highest power of x is 2. These functions have graphs that are called parabolas.
Functions of the form
General form y = x2 + 2x - 3
Factored form y = (x - 1)(x + 3)
Standard form y = (x + 1)2 - 4
In order to get the general form of the quadratic from either of the others, you need only multiply out and simplify, thus this form is also called the expanded form. To get the factored form, you must first have the function in general form, and then factor the expression. And to get the standard form you will want to use the technique of Completing the Square; this too requires the general form as a starting point.
There are two basics shapes for the graph of a quadratic function. In most cases you will be able to deduce the direction of the parabola, i.e. whether it opens up or opens down, by applying the following rule for a quadratic in general form:
|If a is positive, the parabola opens up. See that this function has a minimum y value.|
|If a is negative, the parabola opens down. In this case, observe the function has a maximum y value.|
A particularly important point in the graph of a quadratic is called the vertex. This point is either the maximum or the minimum point of the parabola. If the parabola opens down, the vertex is the maximum, if the parabola opens up, then the vertex is the minimum point. The coordinates of the vertex are most readily seen using the standard form of the quadratic.
The vertex of the quadratic
For example, in the quadratic function we saw above, the standard form is
Justification for the connection between the formula in standard form and the
vertex comes from the
graphing techniques we studied
For the quadratic
If you are given a quadratic function in general form, then to find the vertex you can either rewrite the expression in standard form or else use the following formula.
If the quadratic function is given in general form, i.e.
This formula is derived by rewriting
In the example above,
|An x-intercept is a value at which the graph of the function intersects the x- axis. Algebraically, this means that y = 0.|
Not every quadratic function has 2 x-intercepts. There may be one or even no x-intercepts, as we see below.
y = x2
+ 6x + 9
When is there only one x-intercept?
This happens when the factored form of the quadratic is a perfect square.
Try, for example to factor
||y = x2 + 1|
|When are there no x-intercepts? This will happen when the quadratic is not factorizable via real numbers. That is, we would have to use complex numbers in order to factor it. Try, for example, to factor y = x2 + 1 and see what happens!|
Another option for finding
the x-intercepts is the
quadratic formula. We use it when the function
is given in general form,
|The y-intercept is the value at which the graph of the function intersects the y-axis.|
Every quadratic has a (single) y-intercept.
The reason for this is that the y-intercept is the function
value at x = 0,
and we can always substitute x = 0
into the quadratic. Thus, the y-intercept
of the quadratic function
Now it is time to put your knowledge into practice: