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
y = ax2 + bx + c.
with coefficient a non-zero are called quadratic functions, and their graphs are called parabolas.

Functions of this sort may be written in various ways, depending on our goal in each case. For example, consider the function y = x2 + 2x - 3. It can be written in a number of ways; all of them represent the same function, and therefore all have the same graph.


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.

  • Direction: whether the parabola opens up or down.
  • The vertex of the parabola
  • The x-intercept(s) (there may be 0, 1, or 2 x-intercepts)
  • The y-intercept (there is always one)


These are the four fundamental items of information that we need to know about a parabola in order to sketch it.

Direction

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: y = ax2 + bx + c:

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.

Vertex

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 y = a (x - h)2 + k is the point (h, k).

For example, in the quadratic function we saw above, the standard form is y = (x + 1)2 - 4, so the vertex is at the point (-1, -4).

Justification for the connection between the formula in standard form and the vertex comes from the graphing techniques we studied earlier. For the quadratic y = x2, the vertex is the origin, (0, 0). Subtracting h from x means we have a right horizontal shift by h units if h is positive, or a left horizontal shift by |h| units if h is negative. Adding k to the rest of the expression means we have a vertical shift up by k units if k is positive, or a vertical shift down by |k| units if k is negative. Thus, the new vertex is at (h, k).

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. y = ax2 + bx + c, the x value of the vertex is given by the formula x = -b/2a. The y value of the vertex is found by substituting this into the formula for f (x).

This formula is derived by rewriting y = ax2 + bx + c in standard form.

In the example above, y = x2 + 2x - 3, this gives a vertex with x = -2/2 = -1, and y = (-1)2 + 2(-1) - 3 = 1 - 2 - 3 = -4. Thus, the vertex is again shown to be at (-1, -4).

x-intercepts

An x-intercept is a value at which the graph of the function intersects the x- axis. Algebraically, this means that y = 0.



The most convenient form of the quadratic to use to find the x-intercepts is the factored form; we then set each of the factors equal to 0. In our example above, since the factored form of the function is
y = (x - 1)(x + 3), the x-intercepts are x = 1 and x = -3. Notice that the x value of the vertex is half way between these, as we would expect.

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 + 6x + 9. 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, y = ax2 + bx + c; in this case the x-coordinate of the x-intercepts are given by

y-intercept

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 y = ax2 + bx + c is c. For the other forms of the function, just substitute x = 0 to find the corresponding value of y.

Now it is time to put your knowledge into practice: