The parent function that has positive and negative intervals and crosses through the origin is \( y = \sin x \).
Here's a brief comparison of the functions:
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\( y = \sin x \): This function oscillates between -1 and 1, has positive intervals (between \(0\) and \(\pi\)) and negative intervals (between \(\pi\) and \(2\pi\)), and it crosses through the origin at point \((0, 0)\).
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\( y = x^2 \): This is a parabolic function that opens upwards, meaning it is always positive (except at the origin, where it is zero). It does not have negative intervals.
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\( y = \log x \): This function is only defined for \(x > 0\), so it doesn't cross through the origin and does not exhibit negative intervals.
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\( y = \cos x \): This function oscillates between -1 and 1 and does cross through the y-axis at \( (0, 1) \), but it does not cross through the origin.
Therefore, the answer is \( y = \sin x \).