Question

graph the logarithmic function f(x) = log(x - 1) + 2.

Explanation:

Step1: Find the key points of the parent function \(y = \log x\)

The parent function \(y=\log x\) has a vertical - asymptote at \(x = 0\) and passes through the point \((1,0)\)

Step2: Analyze the transformation for \(y=\log(x - 1)+2\)

For the function \(y=\log(x - 1)+2\), compared to the parent function \(y = \log x\):

• The transformation \(x\to x - 1\) is a horizontal shift to the right by 1 unit. So the vertical asymptote of \(y=\log(x - 1)+2\) is \(x=1\)
• The transformation \(y\to y - 2=\log(x - 1)\) (or \(y=\log(x - 1)+2\)) is a vertical shift up by 2 units.
• When \(x = 2\), \(y=\log(2 - 1)+2=\log(1)+2=0 + 2=2\). So the function \(y=\log(x - 1)+2\) passes through the point \((2,2)\)

Step3: Plot the graph

• Draw the vertical asymptote \(x = 1\) (a dashed line)
• Plot the point \((2,2)\)
• Sketch the curve of the logarithmic function \(y=\log(x - 1)+2\) which approaches the vertical asymptote \(x = 1\) as \(x\to1^{+}\) and increases slowly as \(x\) gets larger

Answer:

To graph \(y=\log(x - 1)+2\), first note the vertical asymptote \(x = 1\). Plot the point \((2,2)\) and sketch the curve approaching \(x = 1\) from the right and increasing as \(x\) increases.