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Standardized tests such as the ACT will not get much more difficult than simply matching a graph to an equation. The equations for sine and cosine wave functions have two **coefficients**, A and B, and are expressed this way:

A{*sin* (Bθ)}

The coefficient A governs the **amplitude** of the graph (how tall it is). The coefficient B governs the **period** of the graph ( how long it takes to complete a cycle).

The equation for the graph below is expressed simply as *y* = sin θ or even simply *y* = sin *x*. The coefficients A and B are both 1.

The equation could be written as *y* = 1(sin)1(θ) or *y* = 1(sin)1(*x*). You can see that the maximum height of the graph is 1 and the minimum is -1; therefore, the amplitude is 1. You can also see that the graph completes a cycle in 2π, so its period is also 1.

You can see that the graph below begins at 1 on the *y* axis, which indicates a cosine wave. The amplitude is 1, and the period is 1.

When the A coefficient increases, the amplitude increases directly.

When the B coefficient increases, the period decreases inversely. So, if B=2, then the period is cut in half. If B=^{1}/_{2}, the period is doubled.

## One More Graph Before We Try Flashcards

See whether you can express the equation for the graph below.

First, determine whether it is sine or cosine. It begins at *y*=0, so it is a sine wave. Next, determine the amplitude. The wave reaches up to 2 and down to -2, so the amplitude is 2.

Next, determine the period. The wave completes a cycle in 4π, so the period is ^{1}/_{2}. Therefore, this graph can be expressed with any of these equations:

*y* = 2 sin ^{1}/_{2}(θ)

*y* = 2 sin ^{1}/_{2}(*x*)

*y* = 2 sin ^{x}/_{2}

## Flashcard Practice

- Is this wave sine or cosine?cosine
- What is the amplitude?2
- What is the period?
^{1}/_{2} - What is the equation for this wave?
*y*= 2 cos^{x}/_{2}

- Is this wave sine or cosine?cosine
- What is the amplitude?2
- What is the period?1
- What is the equation for this wave?
*y*= 2 cos*x*

- Is this wave sine or cosine?cosine
- What is the amplitude?3
- What is the period?2
- What is the equation for this wave?
*y*= 3 cos 2*x*

- Is this wave sine or cosine?sine
- What is the amplitude?3
- What is the period?2
- What is the equation for this wave?
*y*= 3 sin 2*x*

- Is this wave sine or cosine?sine
- What is the amplitude?4
- What is the period?2
- What is the equation for this wave?
*y*= 4 sin 2*x*

### Notice that the next three graphs are SHIFTED UPWARDS!

- What is the period of this wave?1
- What is the amplitude of this wave?2
- Is this wave sine or cosine?sine
- What is the equation for this wave?
*y*= 1 + 2 sin*x*

Notice that this wave completes **3** cycles in 2π.

- What is the period of this wave?3
- What is the amplitude?1
- Is this wave sine or cosine?cosine
- What is the equation for this wave?
*y*= 2 + 3 cos*x*

Notice that this wave completes ** ^{1}/_{2}** cycle in 2π.

- What is the period of this wave?
^{1}/_{2} - What is the amplitude?2
- Is this wave sine or cosine?sine
- What is the equation for this wave?
*y*= 2 + 2 sin^{x}/_{2}