Question

Trig question that has me baffled. y=½(x-2)²-8

Answer 1

In a graph, an intercept means a "touch" or a crossing, namely an x-intercept is when the graph touches the x-axis, thus is called an x-intercept, also goes by the name of the "solutions" or "zeros" or "roots".

when that happens, y = 0.  Likewise, when an y-intercept happens, namely the y-axis gets intercepted by the graph, then x = 0.

to get the x-intercepts, simply set y = 0 and solve for "x".

likewise, to get the y-intercepts, simply set x = 0 and solve for "y".

so let's do so in this case for the x-intercepts,

$\bf y=\cfrac{1}{2}(x-2)^2-8\implies \stackrel{y}{0}=\cfrac{1}{2}(x-2)^2-8\implies 8=\cfrac{(x-2)^2}{2} \\\\\\ 16=(x-2)^2\implies \pm\sqrt{16}=x-2\implies \pm 4= x-2\implies \pm 4 +2 = x$

and you get those two from there, and you know how much that is surely.