Education Technology

Precalculus: Vertical and Phase Shifts

by Texas Instruments


Students explore vertical and phase shifts of sine and cosine functions.

Key Steps

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    Students utilize the Transformation Graphing application to investigate the amplitude of trigonometric functions.

    The program will graph the parent function y = sin(x) and define Y1 = a sin(x) into the screen. Students can enter different values for a and observe the effect on the function. They should find that the sine curve is vertically stretched by a factor of |a|.

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    Next, students investigate the period of a trigonometric graph.

    Here, Y1 = sin(bx), and students substitute given values for b, observing the effects on the function. They will find that the value of b affects the horizontal stretch of this function and thus changes the period.

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    Students investigate a simple phase shift. The program will graph Y1 = sin(x + c) and students substitute given values of c to observe the shift.

    Students then investigate a vertical shift. Investigating as before, students will find that the equation Y1 = sin(x) + d has a vertical shift equal to the parameter d.