Below is an exploration of the Sine and Cosine curves.
In the first picture you will find the parent graph of the two curves.
The equations are-
y= a sin (bx+c) and y= a cos (bx+c), where a=b=1 and c=0
And from here you may join me as I explore addition, multiplication,
division, and compostion of the two functions. And what happens as you
change the a, b, and c values.
In these first set of pictures I am exploring a function that is made by the addition of the two functions.
And how changing the a,b, and c affects each new function.
In this first picture above the purple curve has the equation y= sin x + cos x.
To get to the blue curve and the red curve I changed the a-value. I know from past experience that the 'a' value in a {y= a sin x }equation affects the amplitude( Highest and Lowest points) of the curve. If I treat,y= a sin x+ a cos x , as the new parent equation you can see that changing the a value only affects the amplitude of the parent equation.
Conjecture: All the rules for sine and cosine curves will hold true
for the y= sin x + cos x curve.
In a sin x/cos x graph, where y= sin bx. The 'b' value changes the period of the graph. Essentially either stretching the picture out or squeezing the picture together horizontally.
Below, the blue curve is the parent curve(y= sin bx + cos bx).
For the red curve b=0.5, and in the purple curve b=2
So far, my conjecture holds true.
In a sin x or a cos x curve, where y= sin (x + c). The c-value shifts all the points left or right.
Again in this picture the blue curve is the parent curve(y= sin (x + c) + cos (x +c).
In the red curve c= -pi/2 and in the purple curve c= -pi.
I am pretty sure that my conjecture held true for all a, b, and c when
the two functions were added.
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