# Could you please sketch the following graphs: `y = 2*cos(x-pi/4)` between x = 0 and x = `2*pi`

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### 3 Answers

Sketch the graph of `y=2cos(x-pi/4)` on the interval `[0,2pi]` :

The base graph is `y=cosx` . There is a vertical stretch of factor 2 (The amplitude is doubled) and a horizontal shift of `pi/4` units to the right.

The graph of `y=cosx ("green"),y=2cosx("blue"),y=2cos(x-pi/4)("red")` :

We may graph `y=2cos(x-pi/4) ` using transformation. To do so, consider the graph of the basic function of cosine.

> `y = cos x`

The graph of this within the interval `[0,2pi]` is the blue curve below. Let's take 3 points as our reference `(0,1)` , `(pi/2,0)` and `(pi,-1)` .

Then plot:

> `y = 2cos x`

Multiply the values of y by 2. The reference points become `(0,2)` ,`(pi/2,0)` and `(pi,-2)` . The graph (green curve) is vertically stretch.

Last, plot:

> `y= 2cos(x-pi/4)`

Add the values of x by pi/4. The points become `(pi/4,2)` , `((3pi)/4,0)` and `((5pi)/4,-2)` . So the new graph (red curve) is obtained by shifting `pi/4` units to the right.

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**The red curve is the graph of `y=2cos(x-pi/4)` within the interval `0lt=xlt=2pi` .**

The graph of `y = 2*cos(x - pi/4)` between `x = 0` and `x = 2*pi` is: