Which description matches the graph of $y=-\log (x-2)+3$ (where $\log$ is base 10)?

• A. Vertical asymptote at $x=2$; domain $x>2$; decreasing; passes through $(3,3)$.
• B. Horizontal asymptote at $y=3$; range $y>3$; decreasing; passes through $(3,0)$.
• C. Vertical asymptote at $x=2$; domain $x>2$; increasing; passes through $(3,3)$.
• D. Vertical asymptote at $x=\mathrm{-}2$; domain $x>\mathrm{-}2$; decreasing; passes through $(\mathrm{-}1,3)$.

Answer: (A)

Verified step by step guidance

1. Identify the function given: \(y = -\log(x - 2) + 3\), where \(\log\) is base 10.
2. Determine the vertical asymptote by finding where the argument of the logarithm is zero: set \(x - 2 = 0\), so the vertical asymptote is at \(x = 2\).
3. Find the domain of the function: since the logarithm is defined only for positive arguments, \(x - 2 > 0\), so the domain is \(x > 2\).
4. Analyze the effect of the negative sign in front of the logarithm: the negative sign reflects the graph of \(\log(x - 2)\) across the x-axis, making the function decreasing instead of increasing.
5. Determine the horizontal shift and vertical shift: the \(+3\) shifts the graph up by 3 units, so the function passes through the point \((3, -\log(3 - 2) + 3) = (3, -\log(1) + 3) = (3, 3)\).