For each of the following functions, briefly describe how...

For each of the following functions, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then graph the function using a graphing calculator. Give the domain and the vertical asymptote of each function.
$$y=3+\log _{2} x$$

Key Concepts

Basic Logarithmic Function

A basic logarithmic function, typically in the form y = log_b(x), is a function defined only for positive values of x. Its graph has a characteristic slowly increasing shape, passing through (1, 0), and exhibits important properties such as being continuous and one-to-one. This fundamental function provides the basis for understanding logarithmic behavior and transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, and reflecting the graph of a function. In this case, the function y = 3 + log?(x) is obtained by vertically translating the basic logarithmic graph upward by 3 units. This translation moves every point on the graph the same distance upward without altering the shape of the logarithmic curve.

Domain

The domain of a function is the set of all possible x-values for which the function is defined. For the basic logarithmic function y = log?(x), the domain is x > 0 because logarithms of non-positive numbers are undefined. A vertical translation, such as adding 3 in y = 3 + log?(x), does not affect the domain, so the domain remains x > 0.

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never actually reaches. For the basic logarithmic function y = log?(x), the vertical asymptote is x = 0. Because vertical translations shift the entire graph up or down without changing the position of vertical asymptotes, the graph of y = 3 + log?(x) still has a vertical asymptote at x = 0.

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