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Example: Reading Properties from a Periodic Wavy Graph

By analyzing the visual graph of a mathematical function, various properties can be deduced directly from its plotted coordinate points. Consider a periodic wavy graph extending infinitely in both directions along the horizontal $x$ -axis. Because the curve has a corresponding point for all real $x$ -values, its domain is physically represented as the interval $(-\infty, \infty)$ . Vertically, the function's plotted $y$ -values strictly oscillate between a minimum of $-1$ and a maximum of $1$ , making the corresponding range $[-1, 1]$ . To evaluate specific function values, one must match the given $x$ -input with the precise $y$ -output on the curve. For instance, to evaluate $f\left(\frac{3}{2}\pi\right)$ , locate $\frac{3}{2}\pi$ on the $x$ -axis and read the correlating $y$ -coordinate of the graph, which is $-1$ ; thus, $f\left(\frac{3}{2}\pi\right) = -1$ . Furthermore, to find the $x$ -values where $f(x) = 0$ , identify the $x$ -intercepts where the graph crosses the horizontal axis. For this specific graph, the intercepts appear at points such as $(-2\pi, 0)$ , $(-\pi, 0)$ , (0, 0), $(\pi, 0)$ , and $(2\pi, 0)$ , concluding that the solutions are $-2\pi$ , $-\pi$ , $0$ , $\pi$ , and $2\pi$ . By inspecting the vertical intersection, the single $y$ -intercept occurs where the graph crosses the $y$ -axis when $x = 0$ , which happens precisely at the origin (0, 0).

Updated 2026-06-20

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OpenStax Intermediate Algebra

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