Question

the graph of which function is decreasing over the interval $(-4, \infty)$?\
$\bigcirc$ $f(x) = (x + 4)^2 + 4$\
$\bigcirc$ $f(x) = -(x + 4)^2 + 4$\
$\bigcirc$ $f(x) = (x - 4)^2 - 4$\
$\bigcirc$ $f(x) = -(x - 4)^2 - 4$

Explanation:

Step1: Recall quadratic vertex form

Quadratic functions have the form $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex. If $a>0$, the parabola opens upward; if $a<0$, it opens downward.

Step2: Identify vertex and direction for each option

• Option1: $f(x)=(x+4)^2+4$, $a=1>0$, vertex at $(-4,4)$. Opens upward, increasing on $(-4,\infty)$.
• Option2: $f(x)=-(x+4)^2+4$, $a=-1<0$, vertex at $(-4,4)$. Opens downward, decreasing on $(-4,\infty)$.
• Option3: $f(x)=(x-4)^2-4$, $a=1>0$, vertex at $(4,-4)$. Opens upward, increasing on $(4,\infty)$.
• Option4: $f(x)=-(x-4)^2-4$, $a=-1<0$, vertex at $(4,-4)$. Opens downward, decreasing on $(4,\infty)$.

Step3: Match to required interval

We need the function decreasing on $(-4,\infty)$, which matches Option2.

Answer:

$\boldsymbol{f(x) = -(x + 4)^2 + 4}$