For the graph of `f(x)=-3tan(x)+2`How would I find the period, phase shift, and verticle shift

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txmedteach | High School Teacher | (Level 3) Associate Educator

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To answer this question, we have to see what goes into changing each of these characteristics.

To change the period, we have to shrink or expand the graph in the horizontal direction. Such a horizontal stretch looks like the following conversion:

`f(x) -> f(ax)`

Here, `a` is a constant by which you are multiplying `x`. If `|a|<1`, the graph gets horizontally expanded, and if `|a|>1`, the graph is horizontally shrunk. If you look at the original function you were given, you'll notice that there is no direct multiplier for `x`. Therefore, the period of this function will be the standard period for the tangent function: `pi`.

Now, to find the phase shift, we need to see if any numbers are being added to `x` inside the tangent. A phase shift would look like the following:


The phase shift is a sort of "starting position," where if `x = 0`, the number inside the tangent function is not zero. Clearly, we are adding nothing to the `x` term inside the tangent function, and the phase shift is correspondingly 0

Finally, we must consider the vertical shift. Such a shift would take the following form: 

`tan(x) + c`

Here, `c` tells you how many units to shift the normal graph of `tan(x)` upwards. All we need to do is look at the given equation to see that the upward shift will be 2.