Question

which function has a minimum and is transformed to the right and down from the parent function, $f(x) = x^2$?
$g(x) = -9(x + 1)^2 - 7$
$g(x) = 4(x - 3)^2 + 1$
$g(x) = -3(x - 4)^2 - 6$
$g(x) = 8(x - 3)^2 - 5$

Explanation:

Step1: Identify minimum condition

For a quadratic function $g(x)=a(x-h)^2+k$, if $a>0$, the parabola opens upward and has a minimum. Eliminate options with $a<0$: $g(x)=-9(x+1)^2-7$ and $g(x)=-3(x-4)^2-6$ are discarded.

Step2: Identify right transformation

A horizontal shift right by $h$ units uses $(x-h)$. For right shift, $h>0$. Both remaining options have $(x-3)$, so they shift right 3 units.

Step3: Identify downward transformation

A vertical shift down by $|k|$ units requires $k<0$. Check $k$ values:

• $g(x)=4(x-3)^2+1$ has $k=1$ (shift up)
• $g(x)=8(x-3)^2-5$ has $k=-5$ (shift down)

Answer:

D. $g(x)=8(x-3)^2-5$