what kind of transformation converts the graph of ( f(x)=x^{2}+2 ) into the graph of ( g(x)=9 x^{2}+2 )…

Explanation:

Step1: Recall the transformation rules

For a function (y = f(x)), if we have (y = f(kx)) where (|k|> 1), it is a horizontal shrink. If (y = f(x)) and (y=kf(x)) with (|k| > 1) it is a vertical stretch. The original function is (f(x)=x^{2}+2) and the new function is (g(x)=9x^{2}+2=(3x)^{2}+2). Let (y = f(x)=x^{2}+2) and (y = g(x)). We can rewrite (g(x)) as (g(x)=f(3x)) (since if we substitute (u = 3x) into (f(u)=u^{2}+2), we get (g(x)=(3x)^{2}+2 = 9x^{2}+2)).

Step2: Apply the horizontal - shrink rule

For a function (y = f(x)) and (y = f(kx)) ((k>1)), the transformation is a horizontal shrink. The formula for horizontal transformation: if we start with (y = f(x)) and get (y=f(kx)), the graph of (y = f(x)) is compressed horizontally by a factor of (\frac{1}{k}). Here (k = 3), so the graph of (y=x^{2}+2) is horizontally shrunk to get the graph of (y = 9x^{2}+2).

Answer:

horizontal shrink