A parabola can have three orientations: vertical, horizontal, or inclined. In this post, we will consider the first two orientations of a parabola:
Vertical Orientation
We say a parabola is vertically oriented if its axis is vertical. If we have graph with vertex \((h, k)\), its focus is at \((h, k±a)\). For this type, the standard equation is:
\((x-h)^2=4 a(y-k)\)
The coefficient \(4a\) dictates the opening of the parabola:
- If the value \(4a\) is positive, then we say that the parabola is opening upwards
- If \(4a\) is negative, the graph opens downwards.
If we expand the standard form of a parabola, we get its general form:
\(y=A x^2+B x+C\)
It is a polynomial of the second degree with \(A\), \(B\), and \(C\) as constants.
Horizontal Orientation
We can orient the parabola horizontally if its axis is horizontal. If we have a parabola with vertex \((h, k)\). Its focus is at \((h±a, k)\) and has a standard equation of:
\((y-k)^2=4 a(x-h)\)
Likewise, the coefficient \(4a\) dictates the opening of the parabola:
- If the value \(4a\) is positive, then we say that the parabola is opening to the right
- It opens to the left if \(4a\) is negative.
In general form:
\(x=A y^2+B y+C\)
It is a polynomial of the second degree with \(A\), \(B\), and \(C\) as constants.
Summary
A parabola can have three orientations: vertical, horizontal, or inclined.
We say a parabola is vertically oriented if its axis is vertical. For this type, the standard equation is \((x-h)^2=4 a(y-k)\).
For a vertical parabola: If the value \(4a\) is positive, we say the parabola opens upwards. If \(4a\) is negative, the graph opens downwards.
The general form of a vertical parabola is \(y=A x^2+B x+C\)
We can orient the parabola horizontally if its axis is horizontal. For this type, the standard equation is \((y-k)^2=4 a(x-h)\).
For a horizontal parabola: If the value \(4a\) is positive, then we say the parabola opens to the right. The graph opens to the left if \(4a\) is negative.
The general form of a horizontal parabola is \(x=A y^2+B y+C\)
