Question

1. complete the following.

part a. describe the transformation of $f$ represented by $g$.
$f(x)=\log_{2}x$, $g(x)=\log_{2}(x + 2)-3$
the graph of $g$ is a $\boldsymbol{\square}$ $\boldsymbol{\square}$ units $\boldsymbol{\square}$ and $\boldsymbol{\square}$ units $\boldsymbol{\square}$ of the graph of $f$.

options:
2, 3, 0, 7
up, down, left, right
vertical shift, vertical stretch, horizontal shift, horizontal stretch

Explanation:

Step1: Identify horizontal transformation

For $f(x)=\log_2 x$, $g(x)=\log_2(x+2)-3$ has $x$ replaced by $x+2$. This is a horizontal shift left by 2 units.

Step2: Identify vertical transformation

The $-3$ outside the log function is a vertical shift down by 3 units.

Step3: Combine transformations

The graph of $g$ is a horizontal shift left 2 units and vertical shift down 3 units of $f$.

Answer:

The graph of $g$ is a $\boldsymbol{horizontal\ shift}$ $\boldsymbol{2}$ units $\boldsymbol{left}$ and $\boldsymbol{3}$ units $\boldsymbol{down}$ of the graph of $f$.