Sebastian used the table and correctly identified that the data does not represent a logarithmic function.
What information did Sebastian use in his deduction?
A. The table does not show a vertical asymptote.
B. The table shows two $y$-intercepts and it changes from increasing to decreasing.
C. The table shows one $x$-intercept and one $y$-intercept.
D. The table shows two $x$-intercepts and it changes from increasing to decreasing.
Answer (1)
The data table has points (1, -5), (2, 0), (4, 4), (5, 0), (6, -5).
Logarithmic functions are strictly increasing or decreasing and have at most one x-intercept.
The table shows two x-intercepts (x=2 and x=5) and changes from increasing to decreasing.
Therefore, the correct deduction is: The table shows two x -intercepts and it changes from increasing to decreasing.
Explanation
Analyze the problem We are given a table of x and y values and asked to determine which statement Sebastian used to correctly identify that the data does not represent a logarithmic function. Let's analyze the properties of logarithmic functions and the given data.
Properties of Logarithmic Functions A logarithmic function has a vertical asymptote. The given table does not show any indication of a vertical asymptote, so the first option is plausible. A logarithmic function is strictly increasing or strictly decreasing. The given data changes from decreasing to increasing and then to decreasing again, so it is not a logarithmic function. A logarithmic function has at most one x -intercept. The given data has two x -intercepts at x = 2 and x = 5 . A logarithmic function has one y -intercept. The given data does not have a y -intercept.
Analyze the Options Now let's analyze the given options:
"The table does not show a vertical asymptote." This is true, but not the primary reason why the data is not logarithmic. Logarithmic functions do have vertical asymptotes, but the lack of one isn't the most obvious reason this isn't a logarithmic function.
"The table shows two y -intercepts and it changes from increasing to decreasing." The table does not show two y -intercepts. It doesn't show any y -intercepts (a y -intercept would occur when x = 0 ). Also, while the data does change from increasing to decreasing, the presence of two y -intercepts is incorrect.
"The table shows one x -intercept and one y -intercept." The table shows two x -intercepts (where y = 0 , at x = 2 and x = 5 ) and no y -intercepts. So this is incorrect.
"The table shows two x -intercepts and it changes from increasing to decreasing." The table does show two x -intercepts (at x = 2 and x = 5 ), and the data does change from increasing to decreasing. This is a valid observation that helps to deduce that the data does not represent a logarithmic function.
Final Answer Therefore, the correct answer is: The table shows two x -intercepts and it changes from increasing to decreasing.
Examples
Understanding the characteristics of different types of functions, like logarithmic functions, is crucial in many real-world applications. For example, in analyzing population growth, compound interest, or the decay of radioactive substances, recognizing whether the data fits a logarithmic, exponential, or polynomial model helps in making accurate predictions and informed decisions. In this case, recognizing that the data has two x-intercepts and changes from increasing to decreasing immediately rules out a logarithmic model, which is always increasing or decreasing and has at most one x-intercept.
