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: A logarithmic function \(y = \log_a(x - h)+k\) has a vertical asymptote \(x = h\). The absence of a vertical asymptote in the table data is not the key reason. Logarithmic functions can be transformed, and the lack of a vertical - asymptote in the table might not be conclusive as the table has a limited number of points.
• \(y\) - intercept: The \(y\) - intercept occurs when \(x = 0\). In the given table, there is no \(x = 0\) value, so the claim about two \(y\) - intercepts is wrong.
• \(x\) - intercept: The \(x\) - intercept occurs when \(y=0\). In the table, when \(y = 0\), \(x = 2\) and \(x = 5\) (two \(x\) - intercepts). A logarithmic function \(y=\log_a(x)\) is either always increasing (\(a>1\)) or always decreasing (\(0 < a<1\)). The data in the table first increases (\(y\) from \(- 5\) to \(4\) as \(x\) goes from \(1\) to \(4\)) and then decreases (\(y\) from \(4\) to \(-5\) as \(x\) goes from \(4\) to \(6\)).

Answer:

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