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A vertical shift adjusts the entire graph of a function up or down without changing its shape. For the function \(g(x) = 1 + \log x\), we're starting with the basic logarithmic function \(\log x\). This function is shifted vertically upwards by 1 unit.

To visualize this, imagine taking the graph of \(\log x\) and lifting it so every point is one unit higher on the y-axis. Despite the shift, the overall behavior of the graph remains the same—it rises slowly as \(x\) increases and never crosses the y-axis.

The vertical shift is simply adding a constant to the function, which in our case is the number 1.

• Original function: \(\log x\)
• Shifted function: \(1 + \log x\)
• Effect: Every point on the \(\log x\) graph is now one unit higher

The domain of a function includes all the input values \(x\) for which the function is defined. For logarithmic functions, it's essential to understand what values are allowed.

The function \(\log x\) is defined only for positive values of \(x\), meaning \(x > 0\). This is because logarithms of non-positive numbers are not defined in real number systems.

Similarly, for the function \(g(x) = 1 + \log x\), the domain remains \(x > 0\). The addition of 1 in our function only affects the vertical position of points, not the values \(x\) can take.

• Logarithmic base function domain: \(x > 0\)
• Shifted function domain: \(x > 0\) remains unchanged

Understand the domain is crucial to avoid plotting the graph in areas where it doesn't exist.

Key points are specific values of \(x\) used to help graph a function accurately. They act like anchor points, ensuring that the curve we draw is precise.

For \(g(x) = 1 + \log x\), some useful key points include when \(x = 1\), \(x = 2\), and \(x = 10\). Here's how these are calculated:

• \(x = 1\): \(g(1) = 1 + \log 1 = 1 + 0 = 1\)
• \(x = 2\): \(g(2) = 1 + \log 2 \approx 1.301\)
• \(x = 10\): \(g(10) = 1 + \log 10 = 1 + 1 = 2\)

These key points give us specific coordinates to plot on our graph, like \((1, 1)\), \((2, 1.301)\), and \((10, 2)\). By plotting these coordinates and connecting them with a smooth curve, we ensure our graph reflects the function accurately.

An asymptote is a line that a graph approaches but never actually reaches. In the case of logarithmic functions, these are vertical asymptotes.

For \(\log x\) and our function \(g(x) = 1 + \log x\), there is a vertical asymptote at the line \(x = 0\). The curve will get infinitely close to the y-axis as \(x\) approaches zero from the positive side but will never touch or cross the axis.

Understanding asymptotes is important for comprehending the function's behavior near certain values. Knowing that the function approaches the y-axis without crossing gives us expectations about how to sketch the curve.

• Logarithmic function asymptote: \(x = 0\)
• Behavior: The function nears \(x = 0\) but does not cross it

This characteristic is what helps define the shape and limits of the graph, ensuring that despite any vertical shifts, the asymptotic behavior remains the same.