Question

if you are given the graph of ( h(x)=log _{6} x ), how could you graph ( m(x)=log _{6}(x + 3) )?
translate each point of the graph of ( h(x) ) 3 units up.
translate each point of the graph of ( h(x) ) 3 units down.
translate each point of the graph of ( h(x) ) 3 units right.
translate each point of the graph of ( h(x) ) 3 units left.

Explanation:

For a function \(y = f(x)\), the transformation \(y=f(x + a)\) (where \(a>0\)) is a horizontal translation. The rule is that the graph of \(y = f(x + a)\) is the graph of \(y=f(x)\) shifted \(a\) units to the left. Here, \(h(x)=\log_{6}x\) and \(m(x)=\log_{6}(x + 3)\). Comparing \(y = h(x)\) and \(y=m(x)\), we have \(a = 3\). So, the graph of \(m(x)\) is the graph of \(h(x)\) shifted 3 units to the left.

Answer:

Translate each point of the graph of \(h(x)\) 3 units left.