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How do you sketch the graph of #y=3 cos (x+π) -3#?

Answer 1

I like getting rid of the phase shift (the #x + pi# part) using the sum and difference formulas. The one that is applicable here is

#cos(A + B) = cosAcosB - sinAsinB#.

We have:

#y = 3(cosxcos(pi) - sinxsinpi) - 3#

#y = 3(cosx(-1) - 0) - 3#

#y = -3cosx - 3#

Now you need a little bit of knowledge on the basic cosine function, #y = cosx#. Here's the graph:

graph{y = cosx [-10, 10, -5, 5]}

Whenever there is a coefficient #a# next to the cosine, you have an altered amplitude, which is the distance between the centre (the line #y = 0#) and the top or bottom of the curve.

In the graph of #y = cosx#, the amplitude is simply #1#. In the graph of #y = -3cosx - 3#, the amplitude will be #3#.

The #-# is in front of the #3# to signify a reflection over the x-axis.

Finally, the #-3# to the far right of the equation signifies a vertical transformation of #3# units down. We are left with the following graph:

graph{y = -3cosx - 3 [-10, 10, -5, 5]}

Hopefully this helps!

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.