Question

if $t(x) = x^2$ and $w(x) = t(x - n)$, what is the value of $n$? the graph shows a parabola with points $(-7, 4)$ and $(-3, 4)$ marked, and vertex at $(-5, 0)$ (implied from the symmetry).

Explanation:

Step1: Identify parent function vertex

The parent function $t(x)=x^2$ has vertex at $(0,0)$.

Step2: Identify transformed function vertex

The graph of $w(x)$ has vertex at $(-5,0)$.

Step3: Apply horizontal shift rule

For $w(x)=t(x-n)$, a shift of $t(x)$ to the left by $h$ units gives $t(x+h)$, so $x-n = x+5$.
Solve for $n$:
$x-n = x+5$
$-n = 5$
$n = -5$

Answer:

$-5$