Question

sebastian used the table and correctly identified that the data does not represent a logarithmic function.

x	y
2	0
4	4
5	0
6	-5

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:

1. Analyze logarithmic function properties: Logarithmic functions (e.g., \(y = \log_b(x)\)) have one \(x\)-intercept (when \(y = 0\), \(x = 1\) for basic logs), are either always increasing or always decreasing (depending on the base \(b\)), and have a vertical asymptote at \(x = 0\) (for \(y=\log_b(x)\)).
2. Analyze the table:

• \(x\)-intercepts: Points where \(y = 0\) are \((2, 0)\) and \((5, 0)\) → two \(x\)-intercepts.
• Monotonicity: From \(x = 1\) to \(x = 4\), \(y\) increases (\(-5 \to 0 \to 4\)); from \(x = 4\) to \(x = 6\), \(y\) decreases (\(4 \to 0 \to -5\)) → changes from increasing to decreasing.
• Logarithmic functions are strictly monotonic (always increasing or always decreasing) and have only one \(x\)-intercept. Thus, the table’s two \(x\)-intercepts and non - monotonic (increasing then decreasing) behavior contradict logarithmic function properties.
• Eliminate other options:
• Vertical asymptote: Logarithmic functions have a vertical asymptote, but the table’s lack of showing it is not the key (the data’s behavior is more critical).
• \(y\)-intercept: Only one \(y\)-intercept (\(x = 1\), \(y=-5\) is not a \(y\)-intercept; \(y\)-intercept is at \(x = 0\), not in the table).
• One \(x\)-intercept: The table has two \(x\)-intercepts.

Answer:

D. The table shows two x - intercepts and it changes from increasing to decreasing.