Education Technology

Vertical and Phase Shifts

Published on 01/23/2013

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.