Education Technology

Vertical and Phase Shifts

Activity Overview

Students explore vertical and phase shifts of sine and cosine functions and determine the effect that each change has upon the shape of the graph.

Key Steps

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    First, students review the amplitude and period of a function of the form f(x) = a sin(bx). As students drag the slider controlling the value of a, they will find that the sine curve is vertically stretched by a factor of |a|. Dragging the slider for b, they find that the value of b affects the horizontal stretch of this function and thus changes the period of the function.

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    Next, students explore a vertical shift of the sine function. They will drag the slider for the parameter d of the graph f(x) = sin(x) + d. It will be seen that the vertical shift is equal to this parameter; that is, vertical shift = d.

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    Then, students are given the opportunity to explore a simple phase shift. Student drag the slider for the parameter c in the graph of f(x) = a sin(x+ c). While it is clear that the parameter c affects the horizontal (or phase) shift, it may not be obvious exactly how the number of units of the shift relates to the value of c.