Atticus30idn
Answered

IDNLearn.com offers a user-friendly platform for finding and sharing answers. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.

Which statements are true for the equation [tex]$x^2 = -4y$[/tex]? Check all that apply.

A. The axis of symmetry is [tex]x = 0[/tex].
B. The focus is at [tex](0, -1)[/tex].
C. The parabola opens up.
D. The parabola opens right.
E. The value of [tex]p = -1[/tex].
F. The equation for the directrix is [tex]y = 0[/tex].

Sagot :

GinnyAnsweridn
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.
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.