I love Henning's conic section answer above..... But you can also think about it physically.
A parabola is a good approximation of the path of a physical object that is undergoing constant acceleration. Like a ball thrown on a planet under somewhat constant acceleration from Gravity.
How does this relate to quadratics? First forget about horizontal position.
Imagine as time steps by, we update position and velocity of a particle in one dimension, a particle that you throw down off a cliff on a planet with no atmosphere. y = vertical distance from your position on the cliff. At each time step:
y_position = y_position + y_velocity
y_velocity = y_velocity + y_acceleration
y_acceleration = constant
imagine time steps, with initial position 0, velocity 1, and acceleration of 2
time p v a
0-step: 0, 1, 2
1-step: 1, 3, 2
2-step: 4, 5, 2
3-step: 9, 7, 2
4-step: 16, 9, 2
5-step: 25, 11, 2
6-step: 36, 13, 2
7-step: 49, 15, 2
as u can see, the position is increasing as a square of time, even though we are only using addition to update our particle information. Position turns out to be Time Squared. Position is f(x)=x^2, and time is x.
Why is this quadratic? Because at each step, you add the velocity... but you have previously added the other velocities. At time 4, position is 9+7, but where did the 9 come from? from previously adding 4 and 5. position 16 is actually the sum of all previous velocities - 7 + 5 + 3 + 1 = 16. It becomes sort of like the equation for the sum of numbers between 1 and n, from the famous story about Guass' childhood. That sum is a square. (quadratic). Multiplication is a form of repeated summing, and that's what we are doing here.
What about the non x^2 terms? Like, what if we have f(x) = x^2 + x? Well, the situation turns out to be the same - acceleration is constant. Let's imagine that Position = Time Squared + Time, or f(x) = x^2 + x:
time p v a
0-step: 0+0 , ?, ?
1-step: 1+1 , ?, ?
2-step: 4+2 , ?, ?
3-step: 9+3 , ?, ?
4-step: 16+4, ?, ?
5-step: 25+5, ?, ?
6-step: 36+6, ?, ?
7-step: 49+7, ?, ?
We have no idea what velocity and acceleration are now.... but we can figure them out. First, velocity. Look at the pattern of the position numbers. We know that at each stage, the new position is the old position + velocity. So we can find velocity by looking at the differences.
time p v a
0-step: 0 , 2, ?
1-step: 2 , 4, ?
2-step: 6 , 6, ?
3-step: 12, 8, ?
4-step: 20, 10, ?
5-step: 30, 12, ?
6-step: 42, 14, ?
7-step: 56, ,
We also know acceleration is the difference in velocities across the steps, so we can find a.
time p v a
0-step: 0 , 2, 2
1-step: 2 , 4, 2
2-step: 6 , 6, 2
3-step: 12, 8, 2
4-step: 20, 10, 2
5-step: 30, 12, 2
6-step: 42, 14, 2
7-step: 56, ,
Ohhh... so... by adding in that extra term to our quadratic... f(x)=x^2+x, all we really did was change our initial velocity and our position also changed faster! Acceleration is still constant 2.
All that extra 'x' in the equation did was scale the velocity up in a linear fashion. It didn't make acceleration itself increase.
Adding a constant number to the equation, like f(x) = x^2+x+2 would have a similar result. Acceleration would still be constant.
And if we graph position vs time, we would still get a parabola.
But.... what about the ball thrown with horizontal velocity? Ok... the horizontal speed of the throw doesnt change... the vertical position is what we care about. And the path is a parabola, we are simply stretching time out over a horizontal distance. For a simple example, imagine a 2 dimensional coordinate for a particle, x,y, and imagine it goes like this: x = 0 1 2 3 4, y = 0 1 4 9 16, time = 0 1 2 3 4. f(x) = y = time^2. x = time.
Now if you want to mess around with other stuff and make it more complicated, you can. Time = 2x? Sure. Why not. But the principles are the same. Constant acceleration of the particle produces a quadratic.
This works not only for f(x)=x^2 type situations but for all the ways you can rotate a parabola in space too. (like with the general conic equation linked above)
(Note all this depends on some big assumptions.... for example, that space is everywhere uniform and flat (which is what using Euclidean geometry and Cartesian coordinates imply), gravity is constant (it's not), etc etc. But inverse square laws are found quite often in physics as good approximations of reality.)