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Question 4 • Mathematics III

The parent secant function is shifted 2 units down, and its period is changed to π Which of the following is the graph of the transformed function?

Answer

Image with description On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 2 units and the y axis ranges from negative 2 to 4 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals 3 at (0 comma 3). The graph touches another line y equals 1 at (pi comma 1) and (negative pi comma 1).

Image with description On a coordinate plane, the x axis ranges from negative 2 pi to 2 pi with an interval of pi units and the y axis ranges from negative 4 to 2 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals negative 3 at (2 pi comma negative 1),(0 comma negative 1) and (negative 2 pi comma negative 1). The graph touches another line y equals negative 3 at (pi comma negative 3) and (negative pi comma negative 3).

Image with description On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi units and the y axis ranges from negative 2 to 4 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals 3 at (pi comma 3), (0 comma 3) and (negative pi comma 3). The graph touches another line y equals 1 at (pi by 2 comma 1) and (negative pi by 2 comma 1).

Image with description On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi units and the y axis ranges from negative 4 to 2 with an interval of 1 unit. A secant function with asymptotes at x equals plus or minus n pi is drawn. It touches the line y equals negative 1 at (pi comma negative1), (0 comma negative1) and (negative pi comma negative 1). The graph touches another line y equals negative 3 at (pi by 2 comma negative 3) and (negative pi by 2 comma negative 3).

Explanation

The parent function y = sec(x) has period 2π, so changing its period to π gives y = sec(2x). Shifting it down 2 units gives y = sec(2x) − 2. At x = 0 and x = ±π, the function is 1 − 2 = −1; at x = ±π/2, it is −1 − 2 = −3. Choice B shows these vertices at the correct heights.

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