Equation of a Parabola: Can Any Point on the Graph Satisfy the General Equation?

[kevinshen18]

The general equation of a parabola is:
(y - y1) = A(x - x1)^2

A is the scaling factor
y1 is the y coordinate of a vertex/point on the parabola
x1 is the x coordinate of a vertex/point on the parabola

My two questions are:
1. Can (x,y) be any point on the graph ?
2. If so, then if x - x1 = 0 and y- y1 = 0 (if the difference between the points are 0) then does that mean the point (x,y) is on the parabola?

 

[HallsofIvy]

The general equation of a parabola is:
(y - y1) = A(x - x1)^2

A is the scaling factor
y1 is the y coordinate of a vertex/point on the parabola
x1 is the x coordinate of a vertex/point on the parabola

My two questions are:
1. Can (x,y) be any point on the graph ?

Yes, (x, y) refers to any point on the parabola.

2. If so, then if x - x1 = 0 and y- y1 = 0 (if the difference between the points are 0) then does that mean the point (x,y) is on the parabola?

Your question is a little confusing (or confused). Again, (x, y) can refer to any point on the parabola. "if x- x1= 0 and y- y1= 0" then x= x1 and y= y1 is a specific point on the parabola. In fact, it is the vertex of the parabola- the lowest point if A is positive, highest point if A is negative.

 

[SteamKing]

The general equation of a parabola is:
(y - y1) = A(x - x1)^2

A is the scaling factor
y1 is the y coordinate of a vertex/point on the parabola
x1 is the x coordinate of a vertex/point on the parabola

My two questions are:
1. Can (x,y) be any point on the graph ?

That's what the equations of these curves are for.

Instead of having to compile a list of the infinite number of point coordinates which fall on the curve, a simple equation can be used to generate one point or many points, all of which will fall on the curve

2. If so, then if x - x1 = 0 and y- y1 = 0 (if the difference between the points are 0) then does that mean the point (x,y) is on the parabola?

Any point (x,y) which satisfies the equation of the parabola is a point on that parabola.

Now, why all of this confusion?

 

[Integral]

I would say that (x,y) is some point in the x,y plane, may on the parabola maybe not.

Points on a parabola are given by: (x, (x-x₁)²+ y₁))

 

[HallsofIvy]

In the original post the reference was to (x, y) satisfying the equation $(y- y_1)= A(x- x_1)^2$.

Those (x, y) are points on the parabola, not arbitrary points in the plane.

 

[kevinshen18]

Thanks guys. So if (x,y) = (x1,y1), then this satisfies the parabola equation and (x1,y1) is the vertex?