# What is the amplitude, period and range for ##f(x)=-2sin(2x)-1## and how do you graph it?

The amplitude, or (i.e. how far up and down the graph goes) is always the absolute value of the leading coefficient of the function. In this case, it would be -2. Hence, that is your amplitude. This is because the range of sine (and cosine) functions is [1, -1], and by multiplying the entire function by a certain factor, you’re increasing the output by that factor, changing the overall range.
However, for the range, make sure you consider your starting axis. This is determined by that last constant (in this case -1). This tells you on which ##y=## line the function will start. Hence, you would carry out your amplitude calculations from this point. This would change your range as well. This happens because you’re technically subtracting 1 from the final product of your function, hence making it less than it normally would be. This gives it a vertical shift , and hence a different range.
The period of the graph (i.e. How often it repeats itself) is always the function’s general period divided by the coefficient inside the trig function . No idea what I said? Well, for example, in the above function, what is the coefficient next to ##x##? It’s 2. And the general period of sine (and cosine) functions is ##2pi##. Therefore, the period of this function would be ##(2pi)/2##, or just ##pi##.
The reason this is is because in all sine functions, to complete a period is to get ##2pi## in where ##x## is. By adding the extra coefficient next to ##x##, we’re allowing some leeway for what we need to make ##x## to get that ##2pi##.
Now to graph, you just need to keep these simple tips in mind:

Your graph will be in waves, and these will go up as far as the amplitude.

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Your graph will hit the maximum point every time you complete the period, and will hit the minimum every time you complete half the period.

Your final constant (that -1 in the given) will be like your ##x## axis – you should do all your calculations from there.

Your graph will hit this starting line every quarter-period.

Now all you need to do is keep this stuff in mind, and connect the dots. Here’s what it should look like:
graph{y=-2sin(2x)-1 [-10, 10, -5, 5]}
Hope I didn’t complicate things too much for you ðŸ™‚

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