Question

which is the axis of symmetry of a parabola with equation $y^2 = -8x$? $x = 2$ $x = 0$ $y = -2$ $y = 0$

Explanation:

Step1: Recall parabola standard form

For a parabola of the form \( y^2 = 4ax \) (or \( y^2=-4ax \) for left - opening), the axis of symmetry is the \( x \) - axis (i.e., \( y = 0 \)) when the parabola is symmetric about the \( x \) - axis. The given equation is \( y^{2}=-8x \), which is in the form \( y^{2}=4ax \) where \( 4a=-8\), so \( a = - 2 \).

Step2: Determine axis of symmetry

Since the equation is of the form \( y^{2}=4ax \) (or its variant with a negative sign), the parabola is symmetric about the \( x \) - axis. The equation of the \( x \) - axis is \( y = 0 \). Let's check the other options:

• For \( x = 2 \): This is a vertical line. The parabola \( y^{2}=-8x \) opens to the left, and its vertex is at the origin \((0,0)\). The axis of symmetry is not \( x = 2 \).
• For \( x = 0 \): This is the \( y \) - axis. The parabola \( y^{2}=-8x \) is not symmetric about the \( y \) - axis (if we substitute \( x\) with \(-x\), we get \( y^{2}=8x\), which is a different parabola).
• For \( y=-2 \): This is a horizontal line. The parabola \( y^{2}=-8x \) is not symmetric about \( y=-2\) (if we substitute \( y\) with \(-4 - y\) (to check symmetry about \( y=-2\)), we will not get the same equation).

Answer:

\( y = 0 \)