Phase and vertical shifts

Phase and vertical shifts of sine and cosine

Extend amplitude and period work to general sinusoidal graphs. This precalculus set connects equations, labeled curves and descriptive constraints so learners can identify a midline, a horizontal shift and an appropriate sine or cosine starting point. Factoring the inner coefficient is a central habit throughout.

Practice includes reading and constructing shifted models, finding ranges, locating a cycle’s key points and diagnosing sign or phase errors. Multiselect questions examine equivalent phase descriptions and distinguish a change in vertical position from a change in amplitude or period. A graph-based item requires reading the actual displayed curve rather than relying on a verbal summary.

Before starting, review exact unit-circle values, amplitude and least positive period, and transformations of functions. Angles are in radians, and the standard representation names the sign convention used for a phase shift. Equivalent formulas can describe the same sinusoid; a phase is not always unique without an interval restriction.

The authored practice progresses from parameter interpretation to model recovery and justification. Detailed explanations check extrema, crossing direction, signs and period against the proposed equation. Constant cases are identified explicitly, so a formal coefficient is not mistaken for an observable oscillation. Each question stands independently when presented in a shuffled order.