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When we talk about the amplitude of a sine function, we are discussing how high and low the wave of the function reaches. It’s all about the height of the waves.
The amplitude is determined by the coefficient of the sine term. In the formula like \(y = a \sin x\), the amplitude \(a\) is the absolute value of that coefficient.
For example, let's consider \(y = -4 \sin x\). Here, the coefficient is -4, making the amplitude \(| -4 | = 4\). This means each crest and trough of the wave will extend 4 units up and 4 units down from the center line (x-axis).
Remember, amplitude is always a positive quantity, as it represents distance. Here’s what you need to keep in mind about amplitude:

• It does not depend on whether the coefficient is positive or negative. The negative sign influences other aspects like the direction of the wave, but not the height.


• Amplitude tells you how the wave stretches vertically.

The period of a sine function indicates how long it takes for the function to complete one full cycle. This is its repetition interval. In basic sine functions, this is usually \(2\pi\) radians.
For a standard sine wave like \(y = \sin x\), the period is \(2\pi\), meaning the function repeats every \(2\pi\) along the x-axis.
Let's apply this to our example \(y = -4 \sin x\). Since there are no additional factors attached to \(x\) (like a frequency multiplier), the period remains at \(2\pi\). Here's a quick breakdown:

• For changes in the standard formula \(y = a \sin(bx)\), the period becomes \(\frac{2\pi}{b}\).
• This particular example doesn’t have that \(b\) factor, which is why the period stays \(2\pi\).


• The period determines the length of the x-axis required to see a full wave cycle.

The sine function is one of the fundamental building blocks in trigonometry. It has a characteristic wave-like shape that repeats at consistent intervals.
Graphing the sine function involves understanding its main components: amplitude, period, and any phase shifts. For our specific case, \(y = -4 \sin x\), let's see what these denote:

• **Starting Point:** The graph of \(\sin x\) begins at the origin \((0, 0)\), but a negative sign in front of the coefficient like in \( -4 \sin x \) means it starts descending instead of ascending.
• **Max and Min Values:** Due to the amplitude being 4, the graph will reach a max of 4 and a min of -4. This decides how high and low the curve spikes.
• **Direction:** The negative sign flips the usual sin wave upside down; the curve dips first instead of rising.

To sketch it over one full period \([0, 2\pi]\):

• Start at the origin \((0, 0)\).


• Dip to a low at \(-4\) by \(\frac{\pi}{2}\).


• Return to the midline at \(\pi\).


• Reach a peak at 0 by \(3\frac{\pi}{2}\).


• End at zero, completing the cycle by \(2\pi\).

This path describes one whole wave cycle for the sine function, providing a blueprint for its continuing oscillation.