Sure, let's analyze the given conditions and options step-by-step to determine the correct equation of the parabola.
### Given Conditions:
- The parabola has its vertex at (0,0).
- The focus of the parabola lies along the negative part of the x-axis.
### General Form of a Parabola:
The general forms of parabolas are:
- For horizontal parabolas (opening left/right): [tex]\( y^2 = 4ax \)[/tex]
- Opens right if [tex]\( a > 0 \)[/tex]
- Opens left if [tex]\( a < 0 \)[/tex]
- For vertical parabolas (opening up/down): [tex]\( x^2 = 4ay \)[/tex]
- Opens up if [tex]\( a > 0 \)[/tex]
- Opens down if [tex]\( a < 0 \)[/tex]
Since the focus is along the negative x-axis, the parabola must open leftwards, which implies its equation resembles:
[tex]\[ y^2 = -4ax \][/tex]
where [tex]\( a > 0 \)[/tex].
### Analyzing Options:
1. [tex]\( y^2 = x \)[/tex]:
- This is a horizontal parabola opening to the right.
- Not suitable as it does not match the condition of focusing along the negative x-axis.
2. [tex]\( y^2 = -2x \)[/tex]:
- This is a horizontal parabola opening to the left.
- Suitable as it agrees with the given condition of having the focus along the negative x-axis.
3. [tex]\( x^2 = 4y \)[/tex]:
- This is a vertical parabola opening upwards.
- Not suitable as it focuses along the y-axis rather than the x-axis.
4. [tex]\( x^2 = -6y \)[/tex]:
- This is a vertical parabola opening downwards.
- Not suitable as it focuses along the y-axis rather than the x-axis.
### Conclusion:
The suitable option, considering all conditions and options provided, is:
[tex]\[ \boxed{y^2 = -2x} \][/tex]
