For the following function, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then, graph the function and state the domain and the vertical asymptote of the function.
[tex]f(x)=\frac{1}{4} \log (x-2)-7[/tex]
Describe how the graph of [tex]f(x)[/tex] can be obtained from the graph of a basic logarithmic function.
Shift the graph of [tex]y=\log x[/tex] [$\square$] [$\square$] unit(s), [$\square$] unit(s) [$\square$] and [$\square$]
Use the graphing tool to graph the equation.
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Answer (2)
Shift the graph of y = lo g x to the right by 2 units.
Vertically compress the graph by a factor of 4 1 .
Shift the graph downward by 7 units.
The domain of f ( x ) is ( 2 , ∞ ) , and the vertical asymptote is x = 2 . x = 2
Explanation
Analyzing the Function We are given the function f ( x ) = 4 1 lo g ( x − 2 ) − 7 and we want to describe how its graph can be obtained from the basic logarithmic function y = lo g x . We also need to find the domain and the vertical asymptote of f ( x ) .
Identifying Transformations The function f ( x ) involves several transformations of the basic logarithmic function y = lo g x . Let's identify them one by one:
Horizontal Shift: The term ( x − 2 ) inside the logarithm indicates a horizontal shift. Specifically, it shifts the graph 2 units to the right. This is because we replace x with ( x − 2 ) , so the graph moves to where x − 2 = 0 , i.e., x = 2 .
Vertical Compression: The factor 4 1 in front of the logarithm indicates a vertical compression. It compresses the graph vertically by a factor of 4 1 . This means that the y -values of the graph are multiplied by 4 1 .
Vertical Shift: The term − 7 at the end of the function indicates a vertical shift. It shifts the graph 7 units downward. This is because we subtract 7 from the entire function, so the graph moves down by 7 units.
Describing the Transformations Therefore, to obtain the graph of f ( x ) = 4 1 lo g ( x − 2 ) − 7 from the graph of y = lo g x , we need to:
Shift the graph 2 units to the right.
Vertically compress the graph by a factor of 4 1 .
Shift the graph 7 units downward.
Finding the Domain Now let's determine the domain of f ( x ) . The domain of a logarithmic function is the set of all x values for which the argument of the logarithm is positive. In this case, the argument is ( x − 2 ) , so we need 0"> x − 2 > 0 . Solving this inequality, we get 2"> x > 2 . Therefore, the domain of f ( x ) is ( 2 , ∞ ) .
Finding the Vertical Asymptote Finally, let's find the vertical asymptote of f ( x ) . The vertical asymptote of the basic logarithmic function y = lo g x is x = 0 . Since we shift the graph 2 units to the right, the vertical asymptote of f ( x ) is x = 2 .
Final Answer In summary, the graph of f ( x ) = 4 1 lo g ( x − 2 ) − 7 can be obtained from the graph of y = lo g x by shifting it 2 units to the right, vertically compressing it by a factor of 4 1 , and shifting it 7 units downward. The domain of f ( x ) is ( 2 , ∞ ) , and the vertical asymptote is x = 2 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and determining the pH of a solution. Understanding how to transform logarithmic functions allows us to model and analyze these phenomena more effectively. For example, if we know the population growth of a city follows a logarithmic model, we can use transformations to predict future population sizes based on current trends. Similarly, in seismology, understanding logarithmic scales helps us to compare the magnitudes of different earthquakes and assess their potential impact.
The graph of f ( x ) = 4 1 lo g ( x − 2 ) − 7 can be obtained by shifting the basic logarithmic function 2 units to the right, compressing it vertically by a factor of 4, and shifting down by 7 units. The domain of f ( x ) is ( 2 , ∞ ) and the vertical asymptote is at x = 2 .
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