C2 - Graph transformations (sin, cos, tan)

Hi I don't understand how to draw thw transformations of these graphs.
e.g. cos2x and sin 2x and tan2x

and then what do i do for
2cosx, 2sinx, 2tanx?

I find it extremely confusing :s-smilie:

I know how to draw the graphs for the normal graphs (sinx, cosx, tanx), but have no idea what to do once I have those.

Any help will be much appreciated.
Thank you :smile:

Reply 1

Lunch_Box

Original post by user1-4

Hi I don't understand how to draw thw transformations of these graphs.
e.g. cos2x and sin 2x and tan2x

and then what do i do for
2cosx, 2sinx, 2tanx?

I find it extremely confusing :s-smilie:

I know how to draw the graphs for the normal graphs (sinx, cosx, tanx), but have no idea what to do once I have those.

Any help will be much appreciated.
Thank you :smile:

If y = 2f(x) then this is a y stretch, scale factor 2.

So if y = cosx then y = 2cosx will have its y coordinates doubled like this:

Reply 2

Brister

Original post by user1-4

Hi I don't understand how to draw thw transformations of these graphs.
e.g. cos2x and sin 2x and tan2x

and then what do i do for
2cosx, 2sinx, 2tanx?

I find it extremely confusing :s-smilie:

I know how to draw the graphs for the normal graphs (sinx, cosx, tanx), but have no idea what to do once I have those.

Any help will be much appreciated.
Thank you :smile:

The way I handle sin(2x), cos(2x), tan(2x) etc. is to simply think of it as a change of variable. There is nothing special about the variable x, it is just what we use most often. You could write 2x = z then you would have y = sin(z) and if you plot a graph of y against z, it looks like the sine graph we all know (and love).

However, by convention we draw graphs of y against x. The important thing to remember is that the behavior of the function itself does not change, we just change the way we choose inputs.

For y = sin(x), choosing x = a gives you y = sin(a). If you draw the graph, sin(a) is then the distance from the point (a,0) to the curve.

If instead you have y = sin(2x), when you choose x = a then the value returned from the function is y = sin(2a). To find this you can look again at the curve y = sin(x) and sin(2a) is the distance from (2a,0) to the curve. However, we want the curve y = sin(2x), so we make this the distance of the curve from the point (a,0) instead. The effect is a stretch: for positive x the y-values are moved to the left; for negative x the y-values are moved to the right.

The case where y = 2sin(x) is much simpler, it just means that for any value of x in the domain, the value returned is twice as much as before. Note that another way of thinking about this is that instead of having z = sin(x), we replace z with y/2 which is, again, just a change of variable.

I have attached a diagram which I hope is not too confusing. It shows how points from one curve map onto another. I have only chosen a few specific points, but it should help you see what happens.

Trig Transformations.jpg341.1KB

Reply 3

WhiteGroupMaths

Here's another perspective when sketching sin2x, cos2x and tan2x functions:

Understand that the frequency has doubled as a consequence of the factor 2 multiplier in the above functions when compared to sinx, cosx and tanx.

Hence, say for the sine function, while one cycle was previously completed in a period of 2pi radians as far as sinx is concerned, one cycle will now be completed in pi radians for the sin2x function as the graph is compacted sideways to double the frequency of occurrences.

Hope this helps. Peace.

Reply 4

sghoury

Original post by user1-4

Hi I don't understand how to draw thw transformations of these graphs.
e.g. cos2x and sin 2x and tan2x

and then what do i do for
2cosx, 2sinx, 2tanx?

I find it extremely confusing :s-smilie:

I know how to draw the graphs for the normal graphs (sinx, cosx, tanx), but have no idea what to do once I have those.

Any help will be much appreciated.
Thank you :smile:

The number that preceeds x its all about this x. I assume you know that one cycle of sinx graph is competed in 0-2π. However as the graph is of sin2x , you will now have to divide 2π by 2 which gives π. Now the scale on the y axis remains the same but now one whole cycle of the sinx graph will be completed in π as we calculated, and another cycle will start from π which will end at 2π. so the 2 before x means that there are 2 cycles in this graph!'
Here are the graphs of sinx and sin2x. in the sin2x graph Notice that the values on the y axis remain the same but one whole cycle of the sin graph is being completed in 0-πand the next cycle from π-2π . as calculated by 2π/2=π

Thank you ALL for that. I will probably have to read through it a couple of times before I fully understand it. If I have any further questions I will ask them later.

Thank you!!! :biggrin: