Quadratic Functions
Quadratic functions are among the most familiar in mathematics. The general form of a quadratic function is y = ax2 + bx + c, where a, b, and c are real valued constants, except that a cannot be 0. The graph of a quadratic function is called a parabola, which looks like a "U" with the sides bent outwards. When a is positive, the parabola opens upward, looking somewhat like a smile; when a is negative, the parabola opens downward like a frown. The absolute value of a determines how quickly the parabola rises or falls; the larger the absolute value of a, the more rapidly the parabola rises or falls. The values of b and c in the quadratic equation determine the location of the parabola in the xy-plane. In particular, if b = 0, then the parabola will be symmetric to the y-axis; its highest(or lowest) point, called the vertex, will be located at the point (0,c). If c is also 0, then the vertex will be at the origin, i.e., (0,0). If b is not 0, then the axis of symmetry will no longer be the y-axis; it will be a vertical line with equation x = -b/(2a), so that the first coordinate of the vertex will be -b/(2a). The second coordinate of the vertex also depends on the value of c and may be obtained by substituting -b/(2a) into the function for x.
The simplest of all quadratic functions is y = x2, whose axis of symmetry is the y-axis and whose vertex is (0,0). This is sometimes called the "parent" of all quadratic functions because all other quadratic functions can be generated from it through the use of transformations that scale and translate the graph of y = x2. For example, if we apply the translation which moves the vertex 3 units to the right and 4 units up, we obtain a new parabola with equation y - 4 = (x-3)2, or, with some algebra, y = x2 - 6x + 13 in standard form. Or we could start with y = x2, and scale the second coordinates by a factor of 2, which causes the graph to rise more rapidly for each horizontal increment and which gives the new equation y = 2x2. Or we could do the scaling combined with the translation to obtain the equation y - 4 = 2(x-3)2, which is equivalent to y = 2x2 - 12x + 22 in standard form. In general, if second coordinates of the graph of y = x2 are scaled by a factor of a and the vertex is translated h units horizontally and k units vertically, then we obtain the so-called vertex form for this new parabola, y - k = a(x-h)2. Left in this form, the vertex of the parabola (h,k) is immediately evident. So in some cases, it is convenient to use the vertex form, while in other cases the standard form is preferred. There is a third form of the equation derivable from the geometric definition of a parabola. This definition says that a parabola is the set of all points which are equally distant from a fixed point, called the focus, and a fixed line, called the directrix. If the parabola has vertex at the origin, focus at the point (0,p), and directrix with equation y = -p, then it can be shown that an equation for the parabola is y = (1/(4p))x2. The focus gives the parabola an interesting and useful property. Lines that are directed away from the focus are reflected off the parabola and directed away in lines parallel to the axis of symmetry. Some useful applications of this focal property of parabolas are discussed in the next paragraph.
Quadratic functions have numerous applications in physics and engineering, as well as in other fields. In physics, the quadratic function f(t) = -16t2 is a mathematical model for the freefall of an object due to the earth's gravity when air resistance is negligible. In fact, the flight of a projectile near the earth's surface takes the path of a parabola, the graph of a quadratic function. Engineers use the focal property of parabolas to design searchlights, automobile headlights, telescopes, satellite dishes, and many other devices in which it is necessary to concentrate a beam of light or other electromagnetic signals at a single point. In the case of a searchlight or automobile headlight, the bulb is placed at the focus of a parabolic mirror so that, when it is turned on, the light rays coming from the bulb bounce off the mirror and are all directed as parallel rays away from the mirror, providing a very concentrated beam of light. In the case of a reflecting telescope, light rays coming in from distant objects bounce off the parabolic mirror to the focus where the concentrated light is collected for the astronomer's viewing.
It is often useful to be able to find the point or points at which a quadratic function crosses the x-axis. The first coordinates of these points are called the zeros of the function. This problem is equivalent to solving the equation ax2 + bx + c = 0, whose solution is given by one of the best known formulas in all of mathematics, the quadratic formula. This famous formula gives the zeros in terms of the coefficients a, b, and c. They are x = (-b + (b2 - 4ac) / (2a) and x = (-b - (b2 - 4ac) / (2a). The expression under the radical, b2-4ac, is called the discriminant of the quadratic formula because it discriminates among three possibilities for the zeros of the quadratic function. These possiblities are: (1) if the discriminant is positive, there are two real zeros; (2) if the discriminant is 0, there is only one real zero; and (3) if the discriminant is negative, there are no real zeros, since the square root of a negative quantity is not a real number.
One final interesting fact about parabolas is that they are all similar in a geometric sense, which technically means that any parabola may be mapped onto any other parabola by a similarity transformation. A similarity transformation is some combination of dilations, translations, reflections, and rotations. This means that all parabolas "look alike" in the same sense that all circles look alike or all squares look alike. This may seem counterintuitive when you see the graphs of two different quadratic functions on the same axis system. You might say something like, "one is staying closer to the y-axis than the other one." This gives the illusion that the two parabolas have different shapes. In truth, however, they have the same shape in the same sense that two circles with different radii have the same shape or two squares with sides of different length have the same shape. The proof of this fact is not difficult, but it does require a background in transformations.
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