Question

which is the graph of \\(y = -\log(x - 2) + 3\\)?

Explanation:

Identify the vertical asymptote

Using the Logarithmic Domain knowledge point, the argument of the logarithm must be strictly positive:

$$ x - 2 > 0 \implies x > 2 $$

This means there is a vertical asymptote at \(x = 2\).

Analyze transformations

Using the Logarithmic Transformations knowledge point, we analyze the parent function \(y = \log(x)\):

1. Horizontal shift: \(x \to x - 2\) shifts the graph right by 2 units.
2. Vertical reflection: The negative sign in \(-\log(x-2)\) reflects the graph vertically across the horizontal line of translation.
3. Vertical shift: The \(+3\) shifts the graph upward by 3 units.

Evaluate key points

We calculate specific coordinate points to identify the correct graph:

• For \(x = 3\):

$$ y = -\log_{10}(3 - 2) + 3 = -\log_{10}(1) + 3 = 0 + 3 = 3 $$

So, the point \((3, 3)\) must lie on the graph.

• For \(x = 12\) (assuming base 10):

$$ y = -\log_{10}(12 - 2) + 3 = -\log_{10}(10) + 3 = -1 + 3 = 2 $$

So, the point \((12, 2)\) lies on the graph.

Compare with the given image

The graph shown in the image has:

• A vertical asymptote at \(x = 2\).
• A point at \((3, 3)\).
• A point at \((12, 2)\) (the curve decreases slowly as \(x\) increases).
• As \(x \to 2^+\), \(y \to \infty\) because \(-\log(x-2) \to \infty\).

The graph in the image matches all these characteristics.

Answer:

• (A) The graph shown in the image, which has a vertical asymptote at \(x = 2\), passes through \((3, 3)\), and decreases as \(x\) increases. (Correct answer)
• (B) A graph with a vertical asymptote at \(x = -2\).
• (C) A graph that increases as \(x\) increases from the asymptote \(x = 2\).
• (D) A graph with a vertical asymptote at \(x = 3\).