Let's analyze the equation [tex]\( x^2 = -4y \)[/tex] and determine which statements about it are true.
### Statement 1: The axis of symmetry is [tex]\( x = 0 \)[/tex].
The standard form of a parabola opening up or down is [tex]\( x^2 = 4py \)[/tex]. In our equation [tex]\( x^2 = -4y \)[/tex], the axis of symmetry is the vertical line that passes through the vertex. Here, the vertex is at the origin [tex]\((0, 0)\)[/tex] and the parabola is symmetric about the y-axis. Therefore, the axis of symmetry is indeed [tex]\( x = 0 \)[/tex].
True.
### Statement 2: The focus is at [tex]\((0, -1)\)[/tex].
For the standard form [tex]\( x^2 = 4py \)[/tex], the focus of the parabola is at [tex]\( (0, p) \)[/tex]. To put our equation in the standard form, compare [tex]\( x^2 = -4y \)[/tex] with [tex]\( x^2 = 4py \)[/tex]. Here, [tex]\( 4p = -4 \)[/tex], giving [tex]\( p = -1 \)[/tex]. Therefore, the focus is located at [tex]\( (0, -1) \)[/tex].
True.
### Statement 3: The parabola opens up.
The sign of the coefficient of [tex]\( y \)[/tex] in the equation determines the direction in which the parabola opens. Since the coefficient is negative ([tex]\(-4y\)[/tex]), the parabola opens downward, not upward.
False.
### Statement 4: The parabola opens right.
Similarly, for a parabola that opens left or right, the equation would be in the form [tex]\( y^2 = 4px \)[/tex]. Our equation is [tex]\( x^2 = -4y \)[/tex], meaning it opens either up or down. Due to the negative coefficient, it opens downward, not to the right.
False.
### Statement 5: The value of [tex]\( p = -1 \)[/tex].
From the comparison we did earlier: [tex]\( x^2 = -4y \)[/tex] to [tex]\( x^2 = 4py \)[/tex], we derived [tex]\( 4p = -4 \)[/tex], thus [tex]\( p = -1 \)[/tex].
True.
### Statement 6: The equation for the directrix is [tex]\( y = 0 \)[/tex].
The directrix of a parabola [tex]\( x^2 = 4py \)[/tex] is given by the equation [tex]\( y = -p \)[/tex]. For our equation where [tex]\( p = -1 \)[/tex], the directrix is [tex]\( y = -(-1) = 1 \)[/tex], not [tex]\( y = 0 \)[/tex].
False.
Therefore, the true statements from the list are:
- The axis of symmetry is [tex]\( x = 0 \)[/tex].
- The focus is at [tex]\( (0, -1) \)[/tex].
- The value of [tex]\( p = -1 \)[/tex].
In conclusion, the true statements are 1, 2, and 5.
