Question

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

Explanation:

Step1: Recall parabola standard form

The standard form of a parabola that opens up/down is \(x^{2}=4py\) (opens up if \(p>0\), down if \(p < 0\)). The axis of symmetry is \(x = 0\) (the \(y\)-axis) for this form. The focus is at \((0,p)\) and the directrix is \(y=-p\).
Given equation \(x^{2}=-4y\), compare with \(x^{2}=4py\). So \(4p=-4\), solving for \(p\) gives \(p=- 1\).

Step2: Analyze axis of symmetry

For the parabola \(x^{2}=-4y\), since it is of the form \(x^{2}=4py\), the axis of symmetry is \(x = 0\) (the \(y\)-axis). So the statement "The axis of symmetry is \(x = 0\)" is true.

Step3: Analyze focus

The focus of \(x^{2}=4py\) is \((0,p)\). Since \(p=-1\), the focus is \((0,-1)\). So the statement "The focus is at \((0, - 1)\)" is true.

Step4: Analyze direction of opening

Since \(p=-1<0\), the parabola opens down (because for \(x^{2}=4py\), if \(p < 0\) it opens down, if \(p>0\) it opens up). So "The parabola opens up" is false. Also, the parabola is of the form \(x^{2}=4py\) which opens up/down, not left/right. So "The parabola opens right" is false.

Step5: Analyze value of \(p\)

From \(x^{2}=-4y\) and \(x^{2}=4py\), we have \(4p=-4\), so \(p =- 1\). So the statement "The value of \(p=-1\)" is true.

Step6: Analyze directrix

The directrix of \(x^{2}=4py\) is \(y=-p\). Since \(p=-1\), the directrix is \(y=-(-1)=1\), not \(y = 0\). So "The equation for the directrix is \(y = 0\)" is false.

Answer:

The true statements are:

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