Graphs of sin and cos, how to set points for x values

In summary, the conversation is about graphing trigonometric functions and the process of choosing x values for the graph. The speaker is having trouble understanding the pattern used by the book's authors to choose these values. Another person suggests using the zeros and maximums of cosine to find suitable x values for the graph. The speaker thanks them for the reminder and realizes that they need to equate the angles in parentheses to certain values to find appropriate x values.
  • #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!
 

Attachments

  • Screen Shot 2017-04-23 at 14.16.51.png
    Screen Shot 2017-04-23 at 14.16.51.png
    13.6 KB · Views: 494
  • Screen Shot 2017-04-23 at 14.17.09.png
    Screen Shot 2017-04-23 at 14.17.09.png
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  • #2
A zero of cosine is at π/2. Set the input of cos equal to π/2 and solve for x.
Likewise and a maximum is at 0. Set the input of cos equal to 0 and solve for x.
Those will give you some good x values to use.
 
  • #3
FactChecker said:
A zero of cosine is at π/2. Set the input of cos equal to π/2 and solve for x.
Likewise and a maximum is at 0. Set the input of cos equal to 0 and solve for x.
Those will give you some good x values to use.
Thank you! You reminded me that to find the x values I have to equate angles's values (those in brackets for cos or sin) to 0, π/2, π, etc values to get suitable x values for the graph. :)
 

Related to Graphs of sin and cos, how to set points for x values

1. What is the difference between a sine and cosine graph?

The main difference between a sine and cosine graph is the starting point. A sine graph starts at the origin (0,0) and moves upwards, while a cosine graph starts at its maximum value and moves downwards. Additionally, the shape of the two graphs is slightly different, with the sine graph having a smoother curve and the cosine graph having a sharper curve.

2. How do I determine the x-values for a sine or cosine graph?

The x-values for a sine or cosine graph are determined by the period of the function. The period is the distance between two consecutive peaks or troughs of the graph. To find the x-values, you can use the formula x = k * (2π / b), where k is an integer and b is the coefficient of the x variable.

3. Can I use a calculator to graph a sine or cosine function?

Yes, most scientific calculators have the option to graph trigonometric functions such as sine and cosine. Simply enter the function into the calculator, choose the appropriate window settings, and the graph will be displayed.

4. How do I find the amplitude of a sine or cosine graph?

The amplitude of a sine or cosine graph is the distance from the midpoint of the graph to its maximum or minimum value. It is equal to the coefficient of the trigonometric function, which is usually denoted by the letter "a". For example, if the function is y = 3sin(x), the amplitude is 3.

5. What is the period of a sine or cosine graph?

The period of a sine or cosine graph is the length of one complete cycle of the graph. It is equal to 2π divided by the coefficient of the x variable, which is usually denoted by the letter "b". For example, if the function is y = cos(2x), the period is π.

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