Answer:
The leading coefficient, a.
Explanation:
In a quadratic equation:
y = a*x^2 + b*x + c
The leading coefficient is the coefficient hat that multiplies the variable with the highest power.
In this case, is easy to see that the leading coefficient is a.
We call this the "leading" coefficient because is the coefficient that most impact has on the equation's behavior.
In the case of the quadratic equation, we know that x^2 is always a positive number and is the part that grows faster.
Then if a is a positive number, a*x^2 will be positive, and because this is the part that grows faster, the arms of the parabola will open upwards.
If a is negative, then a*x^2 will be negative, and this will cause that the arms of the parabola open downwards
Then the sign of a defines the direction of the parabola.
And a also defines the width of the parabola.
If a is a really larger number, then
a*x^2 grows really fast
This means that in a small increase of x, we may see a large increase in y, so we will have a "thin" parabola.
On the other hand, if a is really small, a large increase of x will not cause a large increase on y, so we will see a wider parabola.
Then the coefficient a (or variable, depending on which thing are you changing) is the one that controls the parabola's direction and width.