Periodic Functions Revision Notes for Leaving Cert Mathematics

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.

Key Effects

Positive Constant $(k > 0$ )

The graph shifts upward by $k$ units.

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Example: $y = \sin(x) + 2$ moves the sine curve up by 2 units.

Negative Constant ( $k < 0$ )

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

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Example: $y = \cos(x) - 3$ moves the cosine curve down by 3 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.

Worked Examples

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Example 1: Vertical Shift Up

Problem: Graph $y = \sin(x) + 3$

Solution:

The graph of $y = \sin(x)$ is shifted upward by 3 units.
The midline is now $y = 3$

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Example 2: Vertical Shift Down

Problem: Graph $y = \cos(x) - 2$

Solution:

The graph of $y = \cos(x)$ is shifted downward by 2 units.
The midline is now $y = -2$

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.

Periodic Functions (Leaving Cert Mathematics): Revision Notes