Answer :
To analyze the given table and understand why it does not represent a logarithmic function, let's go through the characteristics of the data presented in the table:
\begin{tabular}{|l|l|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & -5 \\
\hline
2 & 0 \\
\hline
4 & 4 \\
\hline
5 & 0 \\
\hline
6 & -5 \\
\hline \hline
\end{tabular}
1. Identifying the Intercepts:
- X-intercepts are the points where the function crosses the x-axis (i.e., \( y = 0 \)). From the table, we can see that \( y = 0 \) when \( x = 2 \) and \( x = 5 \). Therefore, the table shows two x-intercepts: \( x = 2 \) and \( x = 5 \).
- Y-intercepts are the points where the function crosses the y-axis (i.e., \( x = 0 \)). There is no data point in the table for \( x = 0 \), so we cannot identify any y-intercepts from the given data.
2. Behavior of the Function:
- Increasing and Decreasing Intervals:
- From \( x = 1 \) to \( x = 2 \), \( y \) increases from -5 to 0.
- From \( x = 2 \) to \( x = 4 \), \( y \) further increases from 0 to 4.
- From \( x = 4 \) to \( x = 5 \), \( y \) decreases from 4 to 0.
- From \( x = 5 \) to \( x = 6 \), \( y \) further decreases from 0 to -5.
- This indicates that the function is increasing from \( x = 1 \) to \( x = 4 \) and then decreasing from \( x = 4 \) to \( x = 6 \).
3. Characteristics of Logarithmic Functions:
- Logarithmic functions typically have a vertical asymptote (usually on the y-axis when \( x = 0 \)).
- They do not have turning points where the function first increases and then decreases.
- Logarithmic functions typically have only one x-intercept (unless transformed).
Given these observations, let's review the options:
- The table does not show a vertical asymptote: This is true, but it alone is not sufficient to conclude that the function is not logarithmic.
- The table shows two y-intercepts and it changes from increasing to decreasing: This statement is incorrect because the table does not show any y-intercepts.
- The table shows one x-intercept and one y-intercept: This is incorrect because the table shows two x-intercepts and does not show any y-intercepts.
- The table shows two x-intercepts and it changes from increasing to decreasing: This is correct. We have two x-intercepts at \( x = 2 \) and \( x = 5 \), and the function increases from \( x = 1 \) to \( x = 4 \) and then decreases from \( x = 4 \) to \( x = 6 \).
Therefore, based on the information provided in the table, the table shows two x-intercepts and it changes from increasing to decreasing, which is the information that Sebastian used in his deduction that the data does not represent a logarithmic function.
\begin{tabular}{|l|l|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & -5 \\
\hline
2 & 0 \\
\hline
4 & 4 \\
\hline
5 & 0 \\
\hline
6 & -5 \\
\hline \hline
\end{tabular}
1. Identifying the Intercepts:
- X-intercepts are the points where the function crosses the x-axis (i.e., \( y = 0 \)). From the table, we can see that \( y = 0 \) when \( x = 2 \) and \( x = 5 \). Therefore, the table shows two x-intercepts: \( x = 2 \) and \( x = 5 \).
- Y-intercepts are the points where the function crosses the y-axis (i.e., \( x = 0 \)). There is no data point in the table for \( x = 0 \), so we cannot identify any y-intercepts from the given data.
2. Behavior of the Function:
- Increasing and Decreasing Intervals:
- From \( x = 1 \) to \( x = 2 \), \( y \) increases from -5 to 0.
- From \( x = 2 \) to \( x = 4 \), \( y \) further increases from 0 to 4.
- From \( x = 4 \) to \( x = 5 \), \( y \) decreases from 4 to 0.
- From \( x = 5 \) to \( x = 6 \), \( y \) further decreases from 0 to -5.
- This indicates that the function is increasing from \( x = 1 \) to \( x = 4 \) and then decreasing from \( x = 4 \) to \( x = 6 \).
3. Characteristics of Logarithmic Functions:
- Logarithmic functions typically have a vertical asymptote (usually on the y-axis when \( x = 0 \)).
- They do not have turning points where the function first increases and then decreases.
- Logarithmic functions typically have only one x-intercept (unless transformed).
Given these observations, let's review the options:
- The table does not show a vertical asymptote: This is true, but it alone is not sufficient to conclude that the function is not logarithmic.
- The table shows two y-intercepts and it changes from increasing to decreasing: This statement is incorrect because the table does not show any y-intercepts.
- The table shows one x-intercept and one y-intercept: This is incorrect because the table shows two x-intercepts and does not show any y-intercepts.
- The table shows two x-intercepts and it changes from increasing to decreasing: This is correct. We have two x-intercepts at \( x = 2 \) and \( x = 5 \), and the function increases from \( x = 1 \) to \( x = 4 \) and then decreases from \( x = 4 \) to \( x = 6 \).
Therefore, based on the information provided in the table, the table shows two x-intercepts and it changes from increasing to decreasing, which is the information that Sebastian used in his deduction that the data does not represent a logarithmic function.
