Multiple Choice
Graph the given function.
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6. Exponential & Logarithmic Functions
Graphing Logarithmic Functions
Multiple Choice
Which description matches the graph of (where is base 10)?
A
Vertical asymptote at ; domain ; decreasing; passes through .
B
Horizontal asymptote at ; range ; decreasing; passes through .
C
Vertical asymptote at ; domain ; increasing; passes through .
D
Vertical asymptote at ; domain ; decreasing; passes through .
Verified step by step guidance1
Identify the function given: \(y = -\log(x - 2) + 3\), where \(\log\) is base 10.
Determine the vertical asymptote by finding where the argument of the logarithm is zero: set \(x - 2 = 0\), so the vertical asymptote is at \(x = 2\).
Find the domain of the function: since the logarithm is defined only for positive arguments, \(x - 2 > 0\), so the domain is \(x > 2\).
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.
Determine the horizontal shift and vertical shift: the \(+3\) shifts the graph up by 3 units, so the function passes through the point \((3, -\log(3 - 2) + 3) = (3, -\log(1) + 3) = (3, 3)\).
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