Question

what kind of transformation converts the graph of ( f(x)=x^{2}+2 ) into the graph of ( g(x)=9 x^{2}+2 )? vertical stretch horizontal shrink vertical shrink horizontal stretch

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