By analyzing the visual graph of a mathematical function, its properties can be determined directly from the plotted points. Consider a periodic wavy graph that extends infinitely left and right along the $x$-axis. Because it spans all real $x$-values, its domain is $(-\infty, \infty)$. The graph strictly oscillates vertically, reaching its highest points at a $y$-value of $2$ and its lowest points at a $y$-value of $-2$, making its range $[-2, 2]$. To evaluate the function at specific inputs, locate the corresponding point on the curve. For example, to find $f\left(\frac{1}{2}\pi\right)$, identify the point $\left(\frac{1}{2}\pi, 2\right)$ on the graph, which indicates that $f\left(\frac{1}{2}\pi\right) = 2$. Similarly, the point $\left(-\frac{3}{2}\pi, 2\right)$ shows that $f\left(-\frac{3}{2}\pi\right) = 2$. The $x$-intercepts occur where the graph crosses the $x$-axis, such as at $(-2\pi, 0)$, $(-\pi, 0)$, $(0, 0)$, $(\pi, 0)$, and $(2\pi, 0)$; these correspond to the solutions for $f(x) = 0$. The graph crosses the $y$-axis precisely at the origin, meaning the $y$-intercept is $(0, 0)$, which also corresponds to the evaluation $f(0) = 0$.