Throw a ball up in the air, and it flies off in an arc before falling back down — that arc is a parabola. A bridge's arch, the water from a fountain, and the cross-section of a satellite dish are all parabola-shaped too. The equation that represents this kind of curve is the quadratic function, y = ax² + bx + c. The biggest difference from a linear function, whose graph is always a straight line, is that a quadratic function's graph is a smoothly curving arc.
The three constants a, b, and c each play a different role. a decides whether the parabola opens upward (∪ shape) or downward (∩ shape) — positive a gives ∪, negative a gives ∩. The larger |a| is, the narrower and more pointed the parabola gets; the closer to 0, the gentler and wider it spreads out. c decides where the graph crosses the y-axis (the y-intercept), and b affects how far the graph leans left or right. All three values together determine where the parabola's peak — its "vertex" — sits, and the vertex's x-coordinate can always be found with the formula -b/(2a).
One of the most important things to study about quadratic functions is the "x-intercept." This means the point where the graph crosses the x-axis, and the x-value at that point is exactly the same as the solution to the quadratic equation ax²+bx+c=0. In other words, just graphing a quadratic function and looking at where it crosses the x-axis is essentially "solving a quadratic equation by eye." Depending on the sign of the discriminant (b²-4ac), there can be two x-intercepts, one (when the vertex touches the x-axis), or none at all — and seeing it on a graph makes it far more intuitive why these three cases happen.
Parabolas play a central role in physics too. Ignoring air resistance, the path of a thrown object traces a perfect parabola. That's why quadratic functions get used to calculate a cannon's range or predict where a baseball will land. It's also interesting why satellite dishes and flashlight reflectors are shaped as a parabolic cross-section (a paraboloid). A parabola has the property that light or radio waves coming from a single point (the focus) all reflect off and travel out parallel in the same direction — which makes it the ideal shape for gathering or beaming out a signal.
When studying this with kids, it works well to have them change a, b, and c one at a time, in order, and watch how the graph changes at each step. First fix b=0 and c=0 and change only a, to see how the direction it opens and its width change. Then fix a and change only c, to watch the graph move up and down. Finally, change b and watch the vertex shift left and right. This habit of changing one variable at a time turns out to be really useful not just for quadratic functions, but for every multi-variable function you'll study after this.
On our activity page, you can freely adjust the three sliders for a, b, and c and watch the parabola, its vertex, and its x- and y-intercepts change in real time. You can even watch the parabola disappear into a straight line when you set a to 0 — a moment that gives the most vivid possible feel for the condition "a quadratic function requires a to be nonzero."