To solve the given problem and determine which statements are true for the equation [tex]\( x^2 = -4y \)[/tex], we need to analyze the characteristics of the equation of a parabola. Here's a step-by-step breakdown:
1. Axis of Symmetry:
The general form for a vertical parabola is [tex]\( x^2 = 4py \)[/tex]. In this case, the equation [tex]\( x^2 = -4y \)[/tex] can be rewritten as [tex]\( x^2 = 4(-1)y \)[/tex], indicating that [tex]\( 4p = -4 \)[/tex], so [tex]\( p = -1 \)[/tex].
Since the equation is in the form [tex]\( x^2 = 4py \)[/tex], it is symmetric along the y-axis (the vertical line [tex]\( x = 0 \)[/tex]). Therefore, the axis of symmetry is [tex]\( x = 0 \)[/tex].
- True
2. Focus:
For the equation [tex]\( x^2 = 4py \)[/tex], the focus is given by the point [tex]\( (0, p) \)[/tex]. Here, [tex]\( p = -1 \)[/tex], so the focus is [tex]\( (0, -1) \)[/tex].
- True
3. Parabola Opens Up:
In the equation [tex]\( x^2 = 4py \)[/tex], the parabola opens vertically. If [tex]\( p \)[/tex] is positive, the parabola opens up; if [tex]\( p \)[/tex] is negative, the parabola opens down. Given that [tex]\( p = -1 \)[/tex] in this equation, the parabola opens down, not up.
- False
4. Parabola Opens Right:
The form [tex]\( x^2 = 4py \)[/tex] represents a vertical parabola, meaning it does not open horizontally to the right or left. Therefore, the statement that the parabola opens right is incorrect.
- False
5. Value of [tex]\( p \)[/tex]:
As determined earlier, comparing the given equation to the form [tex]\( x^2 = 4py \)[/tex], we find [tex]\( 4p = -4 \)[/tex] which gives [tex]\( p = -1 \)[/tex].
- True
6. Directrix Equation:
The directrix of a parabola [tex]\( x^2 = 4py \)[/tex] is given by the line [tex]\( y = -p \)[/tex]. Here, [tex]\( p = -1 \)[/tex], so the directrix is [tex]\( y = 1 \)[/tex], not [tex]\( y = 0 \)[/tex].
- False
Summarizing, the true statements 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].
Therefore, the results for each statement are:
- The axis of symmetry is [tex]\( x = 0 \)[/tex]. True
- The focus is at [tex]\( (0, -1) \)[/tex]. True
- The parabola opens up. False
- The parabola opens right. False
- The value of [tex]\( p = -1 \)[/tex]. True
- The equation for the directrix is [tex]\( y = 0 \)[/tex]. False
