Mathematics
Quadratic functions
From y = x² to the parabola: the curve behind every throw and many a bridge.
What you need first
Throw a ball forward at an angle and watch it: it rises, keeps slowing as it climbs, tips over the highest point and falls back in an elegant arc. This arc is no random shape, it is a parabola. The same curve is in the water jet of a fountain and in the cables of many a bridge. Behind all of this is one of the most important functions in mathematics: the quadratic function.
y = x²: the basic shape
The simplest quadratic function is y = x². Every x is paired with its square: 2 becomes 4, 3 becomes 9, and −3 also becomes 9, because minus times minus gives plus. If you plot these points, you get a symmetric curve opening upwards: the parabola. Its lowest point is at (0, 0) and is called the vertex. To the left and right of it, the curve rises ever more steeply.
What a and c do
A somewhat more general form is y = a · x² + c with two dials. The number a in front of the x² stretches or squeezes the parabola: with a = 2 every y value is twice as large as for y = x², which makes the curve narrower. If a is negative, as in y = −x², the parabola flips downwards and the vertex becomes the highest point, just like a thrown ball. The number c shifts the whole curve up or down: for y = x² + 3 the vertex is at (0, 3), for y = x² − 5 it is at (0, −5). The fully general quadratic function has one more middle term: y = a · x² + b · x + c. Here we look at the simpler case with b = 0. One thing always holds: a must not be zero, otherwise the x² drops out and all that is left is a straight line.
The vertex
The vertex is the most important point of the parabola: the lowest if it opens upwards, the highest if it opens downwards. In everyday life it often holds the key information: the maximum height of a throw or the lowest point on the cable of a . For y = a · x² + c the vertex is always at (0, c). Often the ground matters just as much: where does the parabola meet the x-axis? There y = 0. For y = 16 − x² this becomes 0 = 16 − x², so x² = 16. Since 4 · 4 = 16, those spots are at x = 4 and x = −4.
Exercises
0 of 6 solvedTime to try it yourself. You can't break anything, every attempt counts.
What is the value of y = x² at x = 3?
Compute y = x² for x = 5.
Compute y = x² for x = −4.
Match each change in y = a · x² + c to its effect on the parabola.
What does the graph of y = −x² look like?
The lowest or highest point of a parabola is called the ….