- #1

Vital

- 108

- 4

## Homework Statement

Hello!

I am at the topic on graphing trigonometric functions. Exercises are rather easy at this point, but I have a problem deciphering how authors of the book choose points for x values. Please, take a look at few examples (including screen shots I attach), and, please, help me to understand the pattern.

## Homework Equations

In all of the following examples the task is to find the period, amplitude, phase shift and vertical shift, and graph at least on cycle. I am making the accent only on x values, as it is easy to compute y values, and there are no problems with those. I would like to understand the pattern for choosing x values.

(1) y = (⅔) cos ( 4x - (π/2) ) + 1

Period: 2π / 4 = π / 2

Amplitude: ⅔

Phase shift: - ( -π/2) / (4) = π / 8

Vertical shift: 1

So to graph the authors have chosen the following values of x:

• π / 8 is chosen as the first x value, because, I presume, we should start the cycle at the point of the phase shift (of course, we can start the cycle anywhere, but here the task is to practice graphs' shifts);

• next x value π/8 + π/8 = π/4;

• next x value π/4 + π/8 = 3π/8;

• next x value 3π/8 + π/8 = π/2;

• next x value π/2 + π/8 = 5π/8;

at this point we finished one cycle with π/2 period, which started at π/8 and finished at 5π/8

Therefore each x value is increased by π/8, which happens to be the value of the phase shift.

(2) y = (-⅓) cos ( (½) x + (π/3) )

Period: 2π / (½) = 4π

Amplitude: 1/3

Phase shift: - ( π/3) / (1/2) = -2π / 3

Vertical shift: 0

So to graph the authors have chosen the following values of x:

• -2π / 3 is chosen as the first x value;

• next x value -2π / 3 + π = π/3;

• next x value π/3 + π = 4π/3;

• next x value 4π/3 + π = 7π/3;

• next x value 7π/3 + π = 10π/3;

at this point we finished one cycle with 4π period, which started at -2π/3 and finished at 10π/3

Therefore each x value is increased by π, which happens to be the value of the phase shift.

## The Attempt at a Solution

I don't see how they choose the pattern which determines the x values.

Thank you very much!