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 (2)
The data has two x-intercepts.
The data changes from increasing to decreasing.
Logarithmic functions are strictly increasing or strictly decreasing and have at most one x-intercept.
Therefore, the table shows two x-intercepts and it changes from increasing to decreasing, which means it is not a logarithmic function. The table shows two x -intercepts and it changes from increasing to decreasing.
Explanation
Analyzing the Problem Let's analyze the given data and the properties of logarithmic functions to determine why Sebastian correctly identified that the data does not represent a logarithmic function.
Identifying Data Points The table contains the following (x, y) data points: (1, -5), (2, 0), (4, 4), (5, 0), (6, -5). We need to determine which characteristic of the data in the table allowed Sebastian to correctly conclude that it does not represent a logarithmic function.
Properties of Logarithmic Functions Let's analyze the properties of logarithmic functions:
Vertical Asymptote: Logarithmic functions have a vertical asymptote. We need to check if the data suggests a vertical asymptote.
Monotonicity: Logarithmic functions are strictly increasing or strictly decreasing. We need to check if the data is strictly increasing or strictly decreasing.
X-intercepts: Logarithmic functions have at most one x-intercept. We need to check how many x-intercepts the data has.
Y-intercepts: Logarithmic functions have at most one y-intercept. We need to check how many y-intercepts the data has.
Observations from the Table From the table, we can observe the following:
The function changes from increasing to decreasing. It increases from (1, -5) to (4, 4) and then decreases from (4, 4) to (6, -5).
The table shows two x-intercepts: at x = 2 and x = 5 (where y = 0).
To find the y-intercept, we would need to find the value of y when x = 0. However, x = 0 is not in the table, so we cannot determine the y-intercept directly from the table. However, logarithmic functions have at most one y-intercept.
Analyzing the Options Now, let's analyze the given options:
The table does not show a vertical asymptote: While this is true, it's not the most direct reason why the data isn't logarithmic. Many functions don't have vertical asymptotes.
The table shows two y-intercepts and it changes from increasing to decreasing: The table does not show two y-intercepts. We cannot determine the y-intercept from the table. However, the function does change from increasing to decreasing, which is a key observation.
The table shows one x-intercept and one y-intercept: The table shows two x-intercepts, not one.
The table shows two x-intercepts and it changes from increasing to decreasing: This statement is correct. The table shows two x-intercepts (x = 2 and x = 5), and the function changes from increasing to decreasing. Logarithmic functions are strictly increasing or strictly decreasing and have at most one x-intercept.
Conclusion Therefore, the information Sebastian used in his deduction is that 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 instance, in analyzing population growth, radioactive decay, or the spread of information, we often use exponential and logarithmic models. Knowing that a logarithmic function is strictly increasing or decreasing and has at most one x-intercept helps us quickly identify whether a given dataset can be modeled using a logarithmic function. This skill is also valuable in fields like finance, where logarithmic scales are used to represent stock prices or investment returns, allowing for a clearer visualization of relative changes.
Sebastian identified that the data does not represent a logarithmic function because it has two x-intercepts and changes from increasing to decreasing. This contradicts the key properties of logarithmic functions, which can only have one x-intercept and must be either strictly increasing or strictly decreasing. Thus, the correct choice is option D: The table shows two x-intercepts and it changes from increasing to decreasing.
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