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How do you graph #y=2cos(2x-pi)+2#?

Answer 1

Samantha Abate

First factor, then pick apart the equation.

Factor out #(2x-pi)# to get #2(x-pi/2)#.

Then, before analyzing the equation, draw the root equation which is #y=cosx#

Now start picking out the important stuff.

Amplitude : a in the equation.

#y=ul2cos(2(x-pi/2))+2#

The amplitude of 2 will stretch #y=cosx# by a factor of 2.

Period : Find the period by looking at the second 2 in the equation.

#y=2cos(ul2(x-pi/2))+2#

Now calculate period: #period=(2pi)/k#

#period=(2pi)/2#

#period= pi#

Apply this to your graph, which will compress it from a period of #2pi# to now just #pi#. (Also can be seen as horizontal compression by factor of #1/2#)

Translations : The last 2 parts of graphing is the translations.

#y=2cos(2(x-ul(pi/2)))+ul2#

The first translation is a horizontal translation right #pi/2# units. So move the graph right #pi/2#. The second translation is a vertical translation up 2 units. Finally, move graph up 2 units.

If you think you may still have it wrong, refer to this handy tool for checking: https://tutor.hix.ai

Voila! All done! Please correct me if I am wrong!

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Answer 2

Josiah Anderson

To graph the function ( y = 2\cos(2x - \pi) + 2 ):

Determine the amplitude, period, phase shift, and vertical shift of the cosine function.
Plot key points using these parameters.
Sketch the graph using these points as a guide.

Amplitude: 2 Period: ( \pi ) Phase Shift: ( \frac{\pi}{2} ) to the right Vertical Shift: 2

Key Points:

Start at the maximum value of the cosine function, which is the amplitude added to the vertical shift: 2 + 2 = 4. At ( x = \frac{\pi}{2} ), the cosine function reaches its maximum value.
At ( x = \pi ), the cosine function reaches its minimum value, which is the vertical shift subtracted from the amplitude: 2 - 2 = 0.
The function repeats every ( \pi ) units to the right, so the next key point will be at ( x = \frac{3\pi}{2} ), where the cosine function reaches its maximum again.
Continue this pattern to sketch the graph.

Plot these key points and sketch the graph accordingly.

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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.