Effect of Adding a Constant to Sin/Cos Function

Overview

When a constant is added to a sine or cosine function, it results in a vertical shift of the graph. The transformation can be represented mathematically as:

$y = \sin(x) + k$ or $y = \cos(x) + k$, where $k$ is a constant.

This vertical shift changes the midline of the graph without affecting the amplitude, period, or frequency of the function.

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Key Effects

Positive Constant $(k > 0$)

The graph shifts upward by $k$ units.

Negative Constant ($k < 0$)

The graph shifts downward by $|k|$ units.

Midline Shift

• The midline of the graph, originally $y=0$, becomes $y=k$
• This new midline serves as the centre of oscillation for the wave.

Unchanged Properties

• Amplitude: Remains the same.
• Period: Remains $2\pi$ for sine and cosine functions.
• Frequency: Remains the same.

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Worked Examples

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Summary

• Vertical Shifts: Adding $k > 0$ moves the graph upward; $k < 0$ moves it downward.
• Midline Change: New midline becomes $y = k$
• Amplitude and Period: No change.
• This transformation adjusts the position of the sine or cosine wave without altering its shape.