Question

9.4.3 quiz: parabolas with vertices at the origin
which of the equations below could be the equation of this parabola?
vertex
(0, 0)
-10
10
a. $x = 4y^{2}$
b. $y = 4x^{2}$
c. $x = -4x^{2}$
d. $y = -4x^{2}$

Explanation:

Step1: Recall the standard form of parabola equations

The standard form of a parabola with vertex at the origin is \( y = ax^{2} \) (opens up or down) or \( x = ay^{2} \) (opens left or right).

Step2: Determine the direction of the parabola

The given parabola opens downwards. For a parabola of the form \( y = ax^{2} \), if \( a<0 \), it opens downwards.

Step3: Analyze each option

• Option A: \( x = 4y^{2} \) is of the form \( x = ay^{2} \) (opens left or right), not the case here.
• Option B: \( y = 4x^{2} \), since \( a = 4>0 \), it opens upwards.
• Option C: \( x=-4x^{2} \) is not a valid parabola equation in standard form (wrong variable on both sides).
• Option D: \( y=-4x^{2} \), here \( a=-4<0 \), and it is of the form \( y = ax^{2} \), which opens downwards.

Answer:

D. \( y = - 4x^{2} \)