Question

which is an equation for the parabola that has vertex (6,-3) and passes through the point (2,-2)? a. $x = -4y^2 - 24y - 42$ b. $x = 4y^2 - 24y + 42$ c. $x = -16y^2 - 48y - 138$ d. $x = -4y^2 - 24y - 30$

Explanation:

Step1: Use vertex form for horizontal parabola

The vertex form of a horizontal parabola is $x = a(y - k)^2 + h$, where $(h,k)$ is the vertex. Here, $h=6$, $k=-3$, so substitute:
$x = a(y + 3)^2 + 6$

Step2: Solve for $a$ using given point

Substitute the point $(2,-2)$ (so $x=2$, $y=-2$) into the equation:
$2 = a(-2 + 3)^2 + 6$
Simplify:
$2 = a(1)^2 + 6$
$2 = a + 6$
$a = 2 - 6 = -4$

Step3: Expand to standard form

Substitute $a=-4$ back into vertex form:
$x = -4(y + 3)^2 + 6$
Expand $(y+3)^2$:
$x = -4(y^2 + 6y + 9) + 6$
Distribute $-4$:
$x = -4y^2 - 24y - 36 + 6$
Combine constants:
$x = -4y^2 - 24y - 30$

Answer:

D. $x = -4y^2 - 24y - 30$