QUADRATIC FUNCTIONS

A function whose values are given by a quadratic polynomial of the form

where a, b, and c are real numbers and c is different from 0, is called a polynomial function of degree 2 or a quadratic function. We shall begin by discussing the simplest form

and constructing a table of values.

x	x^2	y
-3	(-3)^2	9
-2	(-2)^2	4
-1	(-1)^2	1
0	(0)^2	0
1	(1)^2	1
2	(2)^2	4
3	(3)^2	9

Although the table is not complete, it will suffice for beginning to plot point on the coordinate plane.

Once the points have been plotted, it is time to create a smooth curve connecting the points in order of increasing abscissa. The graph of any quadratic function is referred to as a parabola.

It is necessary to point out that upon examining the points, we may choose some x-coordinate, say -2, and its opposite, 2, have the same y-coordinate. Indeed, if a function f has the property that whenever the ordered pair (x,y) belongs to f and the point (-x,y) belongs to f, then the graph of the function y = f(x) is said to be symmetric with respect to the y-axis. From this point on, we shall refer to the y-axis as the axis of symmetry and the function

shall be called the "parent" graph for all quadratic functions.

We should also note that our function has least, or minimum, value. Indeed, the point (0,0) lies below every other point on the graph. This minimum point is called the vertex. The vertex will always be the minimum point if a > 0 and the parabola will open in an upward direction; if a < 0, the vertex will be the maximum point and the parabola will open downward. One way of finding the vertex is to use the formula

Once we have determined the x-coordinate of the vertex, it is rather easy to substitute that value of x back into the original function in order to find the y-coordinate. Consider the function

This function is said to be expressed in quadratic form. It follows that

and

Hence, the vertex is given by (1,-3). We may also view the graph to verify our findings.

Quadratic functions may also be expressed in the form

This is referred to a standard form. A function may be transformed from quadratic form into standard form by completing the square. Let's treat this same example.

The vertex is given by the point (h,k). Notice that our vertex remained (1,-3). Regardless of the form in which the equation is expressed, the vertex is unique.

We shall now try several variations of the parent function, say

and examine their graphs.

Notice that the graph y = f(x) + 1 (in green) and y = f(x)+ 4 (blue) moves one unit up along the y-axis and four units up along the y-axis, respectively. Also, y = f(x) - 2 (in gold) and y = f(x) - 5 (in purple) moves two units down the y-axis and five units down along the y-axis, respectively. It should also be observed that the x-coordinate of the vertex did not change, however the y-coordinate is equal to the value of k.

Let's generalize our results:

If y = f(x) + k and k > 0, the graph is shifted c units up; if k < 0, the graph is shifted c units down.

The vertex is given as the point (0,k).

We should also evaluate our parent function at particular values--such as (x - 2), (x - 3), (x + 1), and (x + 4). Our equations would then be

Now, let's look at their graphs.

We can see that the parent graph is shown in red and our comparisons shall be made in reference to this graph. The graph of y= f(x - 2) is shown in green and the graph has shifted 2 units to the right; y = f(x - 3) is shown in blue and the graph has shifted 3 units to the right; y = f(x + 1) is shown in gold and the graph has shifted 1 unit to the left; y = f(x + 4) is shown in purple and has shifted 4 units to the left. This time, all of the vertices have the same y-coordinate (0), but the x-coordinates are different. In fact, if we choose one of the functions, say y = f(x + 4), then we are able to see that the vertex is given by the point (-4,0).

Let's generalize our results once again:

Consider f(x - h) when y = f(x); if h > 0, the graph undergoes a horizontal shift h units to the right; if h < 0, the graph

undergoes a horizontal shift h units to the left.

The vertex is given by the point (0,h).

So far, we have assumed that the leading coefficient (of the quadratic term) of each function is 1. Let's look at graphs of the function

and examine their graphs for different values of a. Let's begin by considering the values a = 2, a = 3, and a = 4.

Once again, the parent graph is shown in red. The graph given when a = 2 is shown in green; if a = 3, the graph is shown in blue; if a = 4 the graph is shown in gold. It appears as though the larger the integer becomes, the graph stretches vertically more and more. We should also note that although the value of a changed, the vertex did not.

We should also consider negative integers such as a = -2, a = -3, and a = -4.

Notice that these are the same graphs as before; however, they open in a downward direction instead of instead of upward. This is called a reflection about the x-axis. We should also note that the vertex is now the maximum value.

Our investigation should also include values between 0 and 1. Consider the functions

and their graphs along with the parent graph.

The functions are shown in green, blue, and gold, respectively. We should note that as the fraction becomes smaller, the graph shrinks vertically.

We are now able to make several conclusions about the function

If 0 < a < 1, the graph undergoes a vertical shrink.

If a > 1, the graph undergoes a vertical stretch.

If a < 0, the graph is reflected about the x-axis and "behaves" as described above.

SUMMARY

A quadratic function may be expressed in two ways: quadratic form or standard form.

The parabola opens up or opens down--depending on the leading coefficient.

If

1) the vertex is given by the point (h,k);

2) if a > 0, the parabola opens up; if a < 0, the parabola opens down;

3) the axis of symmetry is given by x = h.

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