Sine and cosine graphs, amplitude and period

Sine and cosine graphs, amplitude and period

Learn to connect circular coordinates with the repeating shapes of sine and cosine. This precalculus practice uses exact values, a labeled graph, equations and simple periodic-motion contexts to develop the meaning of amplitude and period. Learners distinguish vertical scale from horizontal scale and recognize how a coefficient changes the number of completed cycles.

Questions cover coordinate projection, graph reading, signed coefficients, zeros over a specified interval, frequency in cycles per unit time and reconstruction of an unshifted model. Conceptual items examine constant functions and explain why a few repeating sample values cannot establish a function’s period everywhere. Both single-select and multiselect formats are included.

Prerequisites are exact unit-circle values and basic function transformations. Angles are in radians unless stated otherwise. Use the graph’s tick labels, track the units in motion problems and distinguish a least positive period from any repeating displacement. No phase or vertical shift is required in this set.

The authored sequence develops from interpretation through computation to model reconstruction and justification. Detailed explanations show how a full turn determines a cycle, how amplitude remains nonnegative and where standard period rules need separate treatment. Shuffled delivery may change the order, but every question contains all the information it needs.