Question

sebastian used the table and correctly identified that the data does not represent a logarithmic function. what information did sebastian use in his deduction? the table does not show a vertical asymptote. the table shows two y - intercepts and it changes from increasing to decreasing. the table shows one x - intercept and one y - intercept. the table shows two x - intercepts and it changes from increasing to decreasing.

Explanation:

• Vertical asymptote: Logarithmic functions \(y = \log_a(x - h)+k\) have a vertical asymptote \(x = h\). The domain of a logarithmic function \(y=\log_a(u)\) (where \(u=x - h\)) is \(u>0\). For the given \(x\) - values (\(x = 1,2,4,5,6\)), if it were a logarithmic function, we would expect a vertical asymptote. But the data does not suggest any such restriction on \(x\) values near a particular vertical line.
• Intercepts and behavior:
• \(x\) - intercept: The \(x\) - intercept is the value of \(x\) when \(y = 0\). From the table, when \(y = 0\), \(x=2\) and \(x = 5\), so there are two \(x\) - intercepts.
• Function behavior: As \(x\) goes from \(1\) to \(4\), \(y\) goes from \(-5\) to \(4\) (increasing). As \(x\) goes from \(4\) to \(6\), \(y\) goes from \(4\) to \(-5\) (decreasing). Logarithmic functions \(y=\log_a(x)\) (\(a>1\)) are always increasing and \(y=\log_{0 < a<1}(x)\) are always decreasing. They do not change from increasing to decreasing behavior.

Answer:

The table shows two \(x\) - intercepts and it changes from increasing to decreasing.