1. The problem is to understand and analyze the function $y=\sin(x^2)$. 2. The function is a composition of the sine function and the square function, meaning we first square $x$ and then take the sine of the result. 3. The sine function, $\sin(\theta)$, oscillates between $-1$ and $1$ for all real numbers $\theta$. 4. Since $x^2$ is always non-negative, the input to the sine function is always $\geq 0$. 5. The function $y=\sin(x^2)$ will oscillate faster as $|x|$ increases because the argument $x^2$ grows quadratically. 6. There are no simple algebraic solutions for zeros other than at $x=0$ where $\sin(0)=0$. Other zeros occur where $x^2 = n\pi$ for integers $n$, so $x=\pm \sqrt{n\pi}$. 7. The function is continuous and smooth everywhere. 8. To graph or analyze further, one might consider derivatives or numerical methods, but the main understanding is the oscillatory behavior with increasing frequency as $|x|$ grows.