Question

which statements are true for the equation x^2 = -4y? check all that apply. the axis of symmetry is x = 0. the focus is at (0, -1). the parabola opens up. the parabola opens right. the value of p = -1. the equation for the directrix is y = 0.

Explanation:

Step1: Recall parabola standard - form

The standard form of a parabola is $x^{2}=4py$. Given $x^{2}=-4y$, we can compare and analyze its properties.

Step2: Find axis of symmetry

For a parabola of the form $x^{2} = 4py$, the axis of symmetry is the line $x = 0$. Since our equation is $x^{2}=-4y$, the axis of symmetry is $x = 0$.

Step3: Determine the value of $p$

Comparing $x^{2}=-4y$ with $x^{2}=4py$, we have $4p=-4$, so $p=-1$.

Step4: Find the focus

The focus of a parabola $x^{2}=4py$ is at the point $(0,p)$. Since $p = - 1$, the focus is at $(0,-1)$.

Step5: Determine the direction of opening

Since $p=-1<0$, the parabola $x^{2}=-4y$ opens down.

Step6: Find the directrix

The equation of the directrix of a parabola $x^{2}=4py$ is $y=-p$. Since $p=-1$, the directrix is $y = 1$.

Answer:

The axis of symmetry is $x = 0$.
The focus is at $(0,-1)$.
The value of $p=-1$.