Which function has a period of pi and is known for vertical asymptotes at multiples of pi/2?

Which function has a period of pi and is known for vertical asymptotes at multiples of pi/2?

• A. Tangent
• B. Sine
• C. Cosine
• D. Quadratic

Answer: (A)

Recognizing a function by its period and where it has vertical asymptotes helps pinpoint the right choice. The tangent function repeats every pi, since tan(x+pi) = tan x. It also has vertical asymptotes where cosine is zero, because tan x = sin x / cos x, so as cos x crosses zero at x = pi/2 + k pi, tan x blows up to infinity. Those x-values are the odd multiples of pi/2, exactly where the asymptotes occur. The sine and cosine functions repeat every 2pi and stay finite everywhere, so they don’t have vertical asymptotes. A quadratic is a simple polynomial with no vertical asymptotes and no fixed period. So the described behavior—period pi and vertical asymptotes at odd multiples of pi/2—fits tangent perfectly.

Recognizing a function by its period and where it has vertical asymptotes helps pinpoint the right choice. The tangent function repeats every pi, since tan(x+pi) = tan x. It also has vertical asymptotes where cosine is zero, because tan x = sin x / cos x, so as cos x crosses zero at x = pi/2 + k pi, tan x blows up to infinity. Those x-values are the odd multiples of pi/2, exactly where the asymptotes occur.

The sine and cosine functions repeat every 2pi and stay finite everywhere, so they don’t have vertical asymptotes. A quadratic is a simple polynomial with no vertical asymptotes and no fixed period. So the described behavior—period pi and vertical asymptotes at odd multiples of pi/2—fits tangent perfectly.