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Algebra Difficulty 3.0 AMC 10/12 Find the answer

The graph of the function y = sin(x) is translated π6\frac{\pi}{6} units to the left, and then each point's abscissa on the graph is changed to 1ω\frac{1}{\omega} (ω > 0) times the original (the ordinate remains unchanged) to obtain the graph of the function y = f(x). If the function y = f(x) has only one zero in the interval (0, π2\frac{\pi}{2}), find the range of ω.

Pick one

Solution

After translating the graph of y = sin(x) π6\frac{\pi}{6} units to the left, we get the graph of y = sin(x+π6x + \frac{\pi}{6}).
Then, changing each point's abscissa to 1ω\frac{1}{\omega} times the original, we obtain the graph of f(x) = sin(ωx+π6\omega x + \frac{\pi}{6}).
In the interval (0, π2\frac{\pi}{2}), ωx+π6(π6,ωπ2+π6)\omega x + \frac{\pi}{6} \in (\frac{\pi}{6}, \frac{\omega \pi}{2} + \frac{\pi}{6} ).
If the function y = f(x) has only one zero in the interval (0, π2\frac{\pi}{2}), then ωπ2+π6(π,2π]\frac{\omega \pi}{2} + \frac{\pi}{6} \in (\pi, 2\pi], hence ω ∈ (53,113](\frac{5}{3}, \frac{11}{3}].

Thus, the answer is: B:(53,113]\boxed{B: (\frac{5}{3}, \frac{11}{3}]}.

This solution uses the properties of the graph transformation rules for the function y = Asin(ωx+φ\omega x + \varphi) and the zeroes of the sine function. The problem primarily assesses understanding of the graph transformation rules for functions of the form y = Asin(ωx+φ\omega x + \varphi) and the zeroes of the sine function, making it a fundamental question.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.