If the parabola is symmetric about x-axis, then we will have square for the variable y. So the given parabola will open upward or downward.
If the parabola is symmetric about y-axis, we will have square for the variable x. So the given parabola will open rightward or leftward.
y2 = 4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.
y2 = -4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.
x2 = 4ay is the standard equation of the parabola which is symmetric about y axis and open upward.
x2 = -4ay is the standard equation of the parabola which is symmetric about y axis and open downward.
Note :
If the given parabola is not in the standard form, then we have to convert it into standard form and decide.
Example 1 :
From the given equation of the parabola, find the direction it opens?
x2 = -16y
Solution :
The given parabola is having square for the variable x, it is symmetric about y-axis.
To decide in which direction does it open, we have to look into the sign. It has negative sign in front of 16y, so the parabola opens downward.
Example 2 :
From the given equation of the parabola, find in which direction it opens?
y2 - 8y - x + 19 = 0
Solution :
Convert the given equation of parabola to standard form.
y2 - 8y = x - 19
y2 - 2y(4) + 42 - 42 = x - 19
(y - 4)2 = x - 19 + 16
(y - 4)2 = x - 3
Let Y = y - 4 and X = x - 3.
Y2 = X
Since the parabola is having square for the variable y, it is symmetric about X-axis.
Since X is positive, the parabola opens to the right.
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