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 Opens to the Right

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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