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.
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
Hence, the vertex is given by (1,-3). We may also view the graph to verify
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
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:
= f(x) + k and k > 0, the graph is shifted c units up; if k < 0, the
graph is shifted c units down.
vertex is given as the point (0,k).
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:
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.
vertex is given by the point (0,h).
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
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
Our investigation should also include values between 0 and 1. Consider the
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
0 < a < 1, the graph undergoes a vertical shrink.
> 1, the graph undergoes a vertical stretch.
< 0, the graph is reflected about the x-axis and "behaves"
as described above.
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.