In order to create the table of points, let's choose some values for x and calculate the corresponding value of y:
[tex]\begin{gathered} x=\frac{1}{2}\colon \\ \log _{\frac{1}{2}}\frac{1}{2}+3=1+3=4 \\ \\ x=1\colon \\ \log _{\frac{1}{2}}1+3=0+3=3 \\ \\ x=2\colon \\ \log _{\frac{1}{2}}2+3=-1+3=2 \\ \\ x=4\colon \\ \log _{\frac{1}{2}}4+3=-2+3=1 \end{gathered}[/tex]
So we have the points (1/2, 4), (1, 3), (2, 2), (4, 1).
Graphing these points and the corresponding curve, we have:
The x-intercept is at x = 8.
As x tends to 0, f(x) tends to infinity.
As x tends to infinity, f(x) tends to minus infinity.
The asymptote is a vertical line at x = 0.
