use the graph to answer the question.\nidentify the logarithmic function shown in the graph.\n(1…

use the graph to answer the question.\nidentify the logarithmic function shown in the graph.\n(1 point)\n$\bigcirc$ $y=-\frac{1}{2}\\log_{2}(x+4)+6$\n$\bigcirc$ $y=-\\log_{2}(x+4)+6$\n$\bigcirc$ $y=-\\log_{2}(x-4)+6$\n$\bigcirc$ $y=-\frac{1}{2}\\log_{2}(x-4)+6$

use the graph to answer the question.\nidentify the logarithmic function shown in the graph.\n(1 point)\n$\bigcirc$ $y=-\frac{1}{2}\\log_{2}(x+4)+6$\n$\bigcirc$ $y=-\\log_{2}(x+4)+6$\n$\bigcirc$ $y=-\\log_{2}(x-4)+6$\n$\bigcirc$ $y=-\frac{1}{2}\\log_{2}(x-4)+6$

Answer

Explanation:

Step1: Find vertical asymptote

The graph has a vertical asymptote at $x=-4$. For a logarithmic function $y=a\log_2(x-h)+k$, the vertical asymptote is $x=h$, so $h=-4$. This eliminates options with $x-4$, leaving $y=-\frac{1}{2}\log_2(x+4)+6$ and $y=-\log_2(x+4)+6$.

Step2: Test point $(0,4)$

Substitute $x=0, y=4$ into $y=-\log_2(x+4)+6$: $y=-\log_2(0+4)+6=-\log_2(4)+6=-2+6=4$. This matches the point.

Step3: Verify with $(-2,6)$

Substitute $x=-2, y=6$ into $y=-\log_2(x+4)+6$: $y=-\log_2(-2+4)+6=-\log_2(2)+6=-1+6=6$. This matches the point.

Answer:

$\boldsymbol{y=-\log_2(x+4)+6}$