Let's carefully examine the symmetry properties of the given equation [tex]\( x^2 + y + 8 = 0 \)[/tex].
### Symmetry with Respect to the x-axis
To determine if the equation is symmetric with respect to the x-axis, we replace [tex]\( y \)[/tex] with [tex]\( -y \)[/tex] in the equation and check if the equation remains unchanged.
1. Original equation:
[tex]\[
x^2 + y + 8 = 0
\][/tex]
2. Substitute [tex]\( y \)[/tex] with [tex]\( -y \)[/tex]:
[tex]\[
x^2 - y + 8 = 0
\][/tex]
Since [tex]\( x^2 + y + 8 \neq x^2 - y + 8 \)[/tex], the equation is not symmetric with respect to the x-axis.
### Symmetry with Respect to the y-axis
To determine if the equation is symmetric with respect to the y-axis, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the equation and check if the equation remains unchanged.
1. Original equation:
[tex]\[
x^2 + y + 8 = 0
\][/tex]
2. Substitute [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]:
[tex]\[
(-x)^2 + y + 8 = 0
\][/tex]
3. Simplify:
[tex]\[
x^2 + y + 8 = 0
\][/tex]
Since [tex]\( x^2 + y + 8 = x^2 + y + 8 \)[/tex], the equation is symmetric with respect to the y-axis.
### Symmetry with Respect to the Origin
To determine if the equation is symmetric with respect to the origin, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] and [tex]\( y \)[/tex] with [tex]\( -y \)[/tex] in the equation and check if the equation remains unchanged.
1. Original equation:
[tex]\[
x^2 + y + 8 = 0
\][/tex]
2. Substitute [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] and [tex]\( y \)[/tex] with [tex]\( -y \)[/tex]:
[tex]\[
(-x)^2 - y + 8 = 0
\][/tex]
3. Simplify:
[tex]\[
x^2 - y + 8 = 0
\][/tex]
Since [tex]\( x^2 + y + 8 \neq x^2 - y + 8 \)[/tex], the equation is not symmetric with respect to the origin.
### Conclusion
The equation [tex]\( x^2 + y + 8 = 0 \)[/tex] is symmetric with respect to the y-axis only.
