2sin(πx-3π)-4

[ArtBog123]

Hi there,

I'm going through Khan Academy problems in Trigonometry and I ran into this one:

2sin(πx-3π)-4

I know 2 is the amplitude, -4 is the midline.
I'm getting hard time understanding "period" in this equation.
Especially the way Khan Academy explains it.

it says:

1. "the transformation from sin(x) to sin(πx) is a horizontal shrunk by a factor of π." - then - "π/2 * 1/π" - Why? why 1/π instead of π/1 ? I understand π/2 is the max point. But π is 1/π, instead of π/1. Please explain that.


2. "the transformation from 2sin(πx) to 2sin(πx-3π)" why it's a translation to the right by 3 units, instead translation to the left by 3 units?

Thanks everyone!

P.S see an attached file.

 

[tkhunny]

It is common to mistake the FORM. ax+b is not good enough. You need a(x-c).

[math]\pi x - 3\pi \rightarrow \pi (x-3)[/math]
[math]\pi[/math] contributes to the period.
[math]3[/math] is the phase shift.

 

[Dr.Peterson]

Shrinking by a factor of pi means division by pi, just as shrinking by a factor of 2 means division by 2. There is some ambiguity in the English, and I have seen sources say this in several different ways. But the meaning is what's important.

To check that meaning, you can note that when x increases by 2, the argument increases by pi times that, which is 2pi. This is the period of the base function sin, so 2 is the period of the new function. Or, using the point they use, when x = 0.5, [MATH]\pi x = \pi/2[/MATH], which is the location of the maximum of the base function.

As for the shift, they explain this, by showing the factored form, as tkhunny mentioned. Again, you can check: When x = 3, the argument is [MATH]\pi x - 3\pi = 0[/MATH], so that point corresponds to the starting point of the base graph.

When the graph is finished, it's a good idea to check everything, by putting the x for some of the key points into the given equation and confirming that you get the y on the graph.

There are probably links to where they explained horizontal stretch and shrink, so you can find deeper explanations of these facts, which they clearly assume you are familiar with. There are also several alternative ways to do each of these steps.