Question

Name 3 of the 4 features listed below for the function g (x) = log2 (x + 4) - 1 and include a description of how you found those answers using complete sentences. 1) Vertical Asymptote 2) Domain 3) X and Y Intercepts 4) Transformations compared to its parent function f (x) = log2 x

Answer 1

(1) Vertical asymptote: $x=-4$

(2) Domain: $x>-4$

(3) X intercept: $(-2,0)$ and Y intercept : $(0,1)$

(4) The function g(x) is shifted 4 units to the left and shifted 1 unit down.

Explanation:

The parent function is $f(x)=\log _{2} x$

The transformed function is $g(x)=\log _{2}(x+4)-1$

(1) Vertical asymptote:

The vertical asymptote of a function can be determined by equating

$x+4=0$

Thus, $x=-4$

The vertical asymptote is $x=-4$

(2) Domain:

The domain of a function is the set of all independent x-values.

$x+4>0$

Thus, $x>-4$

The domain of a function is $x>-4$

(3) X and Y intercepts:

To determine the x intercept, let us substitute y=0 in $g(x)=\log _{2}(x+4)-1$

$\begin{equation}\begin{aligned}\log _{2}(x+4)-1 &=0 \\\log _{2}(x+4) &=1 \\x+4 &=2^{1} \\x &=-2\end{aligned}$

Thus, the x intercept is $(-2,0)$

To determine the y intercept, let us substitute x=0 in $g(x)=\log _{2}(x+4)-1$

$\begin{equation}\begin{aligned}y &=\log _{2}(0+4)-1 \\&=\log _{2} 4-1 \\&=2-1 \\&=1\end{aligned}$

Thus, the y intercept is $(0,1)$

(4) To determine the transformation:

The transformed function $g(x)=\log _{2}(x+4)-1$ is shifted 4 units to the left and shifted 1 unit downwards.