Read it again any time, including while you work through the exercises.
Throw a ball toward a hoop and watch its path: it rises, peaks, then comes back down, tracing a rounded and perfectly symmetric curve. This curve is called a parabola, and it is the graphic signature of quadratics, that is, functions where x appears squared.
The basic model is the square function, f(x) = x². Compute a few values: f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4. Two things stand out. First, a number and its opposite have the same square, so the curve is symmetric about the vertical axis. Second, the values go down to 0 then climb back up: f is decreasing on ]-∞, 0], increasing on [0, +∞[, with a minimum of 0 reached at 0.
Every function of the form f(x) = ax² + bx + c, with a nonzero, keeps this parabola shape. Its lowest or highest point is called the vertex, and the vertical line through the vertex is the axis of symmetry: the parabola folds exactly onto itself there. The x-coordinate of the vertex comes from a formula worth knowing by heart: x = -b/(2a). To find the y-coordinate of the vertex, you then simply compute the image of that value.
The sign of a decides the orientation. If a > 0, the parabola opens upward, like a bowl: it goes down then up, and the vertex is a minimum. If a < 0, it opens downward, like a bell: it goes up then down, and the vertex is a maximum. Vertex, axis of symmetry, orientation: with these three keys, you can sketch any parabola in seconds.