Sinusoidal graphs (y=sinx and y=cosx equations)

[becca_j]

i was wondering how you would find the amplitude, vertical translation, horizontal translation and the sinusoidal axis either from just an equation, like 1/2(y-7)=cos3(x-60), or from the graph if you are given one.

 

[mathman]

Write it in the form y= and the items you want would be obvious.

amplitude = 2, vertical (also axis) = 7, period = 2pi/3

 

[tacman]

To find the amplitude of a sinusoidal graph, you need to look at the coefficient of the sine or cosine function. In the equation 1/2(y-7)=cos3(x-60), the coefficient of the cosine function is 1/2. This means that the amplitude is 1/2.

To find the vertical translation, you need to look at the constant term in the equation. In this case, the constant term is -7. This means that the graph is shifted down 7 units.

To find the horizontal translation, you need to look at the value inside the parentheses. In this case, it is (x-60). This means that the graph is shifted to the right 60 units.

The sinusoidal axis is the midline of the graph and can be found by taking the average of the maximum and minimum values of the graph. If you are given a graph, you can simply draw a horizontal line passing through the middle of the graph and determine the y-value at that point. This will give you the sinusoidal axis.

If you are given an equation and asked to graph it, you can use the information above to plot points and create the graph. Alternatively, you can also use transformations to graph the equation. For example, for the equation y=cosx, the graph has an amplitude of 1, no vertical or horizontal translation, and a sinusoidal axis at y=0. This means that the graph will have a maximum value of 1 and a minimum value of -1, with the x-axis as the midline. To graph y=cos3x, you can use the following transformations:

- The coefficient of x is multiplied by 3, which means the graph will be compressed horizontally by a factor of 3.
- The graph is not shifted vertically or horizontally, so there is no vertical or horizontal translation.
- The sinusoidal axis remains at y=0.

Using these transformations, you can plot points and create the graph of y=cos3x. Similarly, you can use transformations to graph any sinusoidal equation given to you.