To determine the equation of a parabola based on the given conditions, let's analyze the options given.
Key details:
- The vertex of the parabola is at [tex]\((0, 0)\)[/tex].
- The focus of the parabola is along the negative part of the [tex]\(x\)[/tex]-axis.
For a parabola with its vertex at the origin [tex]\( (0, 0) \)[/tex] and its focus on the negative [tex]\( x \)[/tex]-axis, the general form of the equation is:
[tex]\[ y^2 = -4px \][/tex]
where [tex]\( p \)[/tex] is the distance from the vertex to the focus. Since the focus lies on the negative [tex]\( x \)[/tex]-axis, [tex]\( p \)[/tex] is a positive value (because we're moving left from the origin).
Now, let’s look at the provided options and check which one of them matches the form [tex]\( y^2 = -4px \)[/tex]:
1. [tex]\( y^2 = x \)[/tex]
- This form does not match [tex]\( y^2 = -4px \)[/tex], as there is no negative coefficient for [tex]\( x \)[/tex].
2. [tex]\( y^2 = -2x \)[/tex]
- This equation is of the form [tex]\( y^2 = -2x \)[/tex], which suggests it has a negative coefficient.
- We can rewrite it as [tex]\( y^2 = -2x \)[/tex], which fits the form [tex]\( y^2 = -4px \)[/tex] with [tex]\( 4p = 2 \)[/tex] or [tex]\( p = \frac{1}{2} \)[/tex], indicating that the focus would be at [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex].
- This matches our requirement that the focus is on the negative [tex]\( x \)[/tex]-axis.
3. [tex]\( x^2 = 4y \)[/tex]
- This form represents a parabola that opens either upwards or downwards, which is not what we need since the focus should be on the [tex]\( x \)[/tex]-axis.
4. [tex]\( x^2 = -6y \)[/tex]
- This form represents a parabola that opens downwards, which does not fit our criteria of the focus being on the [tex]\( x \)[/tex]-axis.
After examining the given options, the suitable equation for a parabola with a vertex at [tex]\((0,0)\)[/tex] and its focus along the negative [tex]\( x \)[/tex]-axis is [tex]\( y^2 = -2x \)[/tex].
Hence, the correct answer is:
[tex]\[ y^2 = -2x \][/tex]
