1. **Problem Statement:** We need to sketch the graph of the function $$y=3\sin\left(x^2\right)$$ for the domain $$0 \leq x \leq 2\pi$$. 2. **Formula and Explanation:** The function is a sine function with the argument squared, scaled by 3. The general sine function is $$y=\sin(x)$$ which oscillates between -1 and 1. Multiplying by 3 scales the amplitude to oscillate between -3 and 3. 3. **Key Points:** - The input to sine is $$x^2$$, so the frequency changes non-linearly. - At $$x=0$$, $$y=3\sin(0)=0$$. - At $$x=\sqrt{\frac{\pi}{2}}$$, $$y=3\sin\left(\frac{\pi}{2}\right)=3$$ (maximum). - At $$x=\sqrt{\pi}$$, $$y=3\sin(\pi)=0$$. - At $$x=\sqrt{\frac{3\pi}{2}}$$, $$y=3\sin\left(\frac{3\pi}{2}\right)=-3$$ (minimum). - At $$x=\sqrt{2\pi}$$, $$y=3\sin(2\pi)=0$$. 4. **Behavior:** The sine argument grows faster than linearly, so oscillations get closer as $$x$$ increases. 5. **Summary:** The graph oscillates between -3 and 3 with increasing frequency as $$x$$ goes from 0 to $$2\pi$$. Final answer: The function is $$y=3\sin\left(x^2\right)$$ for $$0 \leq x \leq 2\pi$$, with amplitude 3 and non-linear frequency due to the squared argument.