Compare the parent functions in the following answer choices. Which parent function has positive and negative intervals and crosses through the origin?

Question

1)y=sin x
2)y=x^2
3)y=cos x
4)y=log x

Answer 1

To determine which parent function has positive and negative intervals and crosses through the origin, let's analyze each option:

1. \(y = \sin x\): The sine function oscillates between -1 and 1. It is negative on intervals like \((\pi, 2\pi)\) and positive on intervals like \((0, \pi)\). It crosses through the origin at \(x = 0\).

2. \(y = x^2\): This is a parabola that opens upwards. It is always non-negative (i.e., \(y \geq 0\)), being zero only at the origin. Thus, it does not have negative intervals.

3. \(y = \cos x\): The cosine function also oscillates between -1 and 1. It is negative on intervals like \((\frac{\pi}{2}, \frac{3\pi}{2})\) and positive on intervals like \((0, \frac{\pi}{2})\). However, it does not cross through the origin since \(\cos(0) = 1\).

4. \(y = \log x\): The logarithmic function is only defined for \(x > 0\) and is always positive in the interval \((1, \infty)\) and negative in the interval \((0, 1)\). It does not cross through the origin, as \(\log(1) = 0\) and it is undefined for \(x \leq 0\).

Based on this analysis, the only parent function that has both positive and negative intervals and crosses through the origin is:

1) \(y = \sin x\).