To examine the graphs of y= a sin (bx + c ). Let begin by examining the graph of y = sin x and then add the parameters a, b, and c to the graph one at a time. Finally, in conclusion we will put all three parameters together and then describe the movement in the position and the shape of the graph

y= a sin ( bx + c ).

Click on the equation to see the graph.

y= sin x

Now consider the possible values for a. If a = 0, then there is nothing to consider. In the graph of y = sin x the value of a is one. A logical question is what if a = -1 ? Click the equation to see the graph.

y = - sin x

What did you notice about the graph?

My observations.

Make a conjecture about the graph y= a sin x if -10 < a < 10. Click on the equation to see an animation of the graph for the values of a. On the graph let a = n so that the equation editor can produce the animation. Click the equation to see the graph.

y = a sin x

Did what you think would happen, happen?

My observations.

Now lets see what happen when we hold a constant at 1 and c constant at 0, and see what effect b has on the graph of y = sin bx. What do you get if b = 0? Make a conjecture and click the equation to see the graph.

y = sin 0x

My observations.

Next consider values b which are greater than zero. Make a conjecture and click the equation to see the graph. You can stop the animation by clicking on the play buttons. The animation is for 0 < b < 10.

y = sin bx

Was your conjecture correct? Did you see the effect of b on the sine curve? If not, click on the eqaution below to see graphs for specific values of b.

y = sin bx

Now consider values of b which are less than zero. Make a conjecture and click the equation to see the graph. You can stop the animation by clicking on the play buttons. The animation is for -10 < b < 0.

y = sin bx

Was your conjecture accurate? Did you see the effect of decreasing b?If not, click on the eqaution below to see graphs for specific values of b.

y = sin bx

My observations.

To investigate the effects of the parameter c on the graph of y = sin x we should hold a and b constant at1. If c = 0 there is nothing to consider. Make a conjecture about the graph of y = sin (x + c ) for c > 0. Click the equation to see the graph. You can stop the animation by clicking on the play buttons. The animation is for 0 < c < 10.

y = sin ( x + c )

Can you see what is happening? If you would like to see the graph for specific values of c, click the equation below.

y = sin ( x + c )

Next consider values of c that are less than zero. Make a conjecture about the graph of y = sin (x + c ) for c < 0. Click the equation to see the graph.

y = sin ( x + c )

Could you see what was happening as c decreased from 0 to -10? f you would like to see the graph for specific values of c, click the equation below.

y= sin ( x + c)

My observations.

Lets conclude by putting all the parameters back into the equation and see what happens. Amagine what the graph of y= 2 sin (2x +3) would look like compared to the graph of y = sin x (here a=1, b=1, and c=0). Click on the equation to see the graphs. By clicking on a color for each equation in the equation menu you can add each parameter one at a time to see if you have identified the effects.

y= sin x

If you understand this demonstration then you should be able to write and equation for a sine wave in your are given the graph. If you would like to put your newly aquired knowlege to the test write and equation for this graph. Don't look at the equation at the top of the page until you have written your equation.