Question

what kind of transformation converts the graph of $f(x) = -x - 1$ into the graph of $g(x) = -6x - 6$?
horizontal stretch
horizontal shrink
vertical shrink
vertical stretch

Explanation:

Step1: Analyze the relationship between \(f(x)\) and \(g(x)\)

We know that \(g(x)=-6x - 6\) and \(f(x)=-x - 1\). We can rewrite \(g(x)\) as \(g(x)=6(-x - 1)\).
Since \(g(x)=6f(x)\) (because if \(y = f(x)=-x - 1\), then \(y'=g(x)=6(-x - 1)\)).

Step2: Recall the rules of vertical transformation

For a function \(y = f(x)\), if \(y'=af(x)\) (\(a> 1\)), the transformation is a vertical stretch. Here \(a = 6>1\).

Answer:

vertical stretch