IDENTIFY THE DIRECTION A PARABOLA OPENS
About "Identify the direction a parabola opens"
Identify the direction a parabola opens :
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.
Parabola 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.
Parabola symmetric about x-axis and open leftward
y² = -4ax is the standard equation of the parabola which is symmetric about x axis and open rightward.
Parabola symmetric about y-axis and open upward
x² = 4ay is the standard equation of the parabola which is symmetric about y axis and open upward.
Parabola symmetric about y-axis and open downward
x² = -4ay is the standard equation of the parabola which is symmetric about y axis and open downward.
Let us look into some example problems to understand the above concept.
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 in which direction does it open?
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 does it open?
y² - 8y - x + 19 = 0
Solution :
Let us convert the given question in 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.
It has positive sign, so the parabola opens rightward.
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