when you have "y = 1/2sin2x". . .

Does the equation have a vertical compression by a factor of 1/2?

And does the equation have a horizontal compression by a factor of 2?

Well, we could argue about semantics, but I agree with what you are trying to say.

In both cases, it is squeezed by a factor of 2. In the first case y dimensions are halved. In the second case x dimension are halved.

Now if you want to express this as a scale change transformation.

It would be
S1,1/2 for the vertical shrink
and
S1/2,1 for the horizontal shrink

In the vertical case your scale change matrix operation would look like this

|1 0 | x = | 1 x|
|0 .5| y |1/2 y|

In the horizontal shrink, like this
|.5 0| x |1/2 x|
|0 1 | y = |1 y |

Now if you do them both at once, as you did, the Scale is S1/2,1/2

and the matrix is
|.5 0|
|0 .5|

To determine whether the equation "y = (1/2)sin(2x)" has a vertical compression by a factor of 1/2 and a horizontal compression by a factor of 2, we need to understand the effects of these factors on the graph of the sine function.

1. Vertical compression by a factor of 1/2:
When we have a value less than 1 as the coefficient of the sine function, it results in a vertical compression of the graph. In this case, the coefficient is (1/2), which means the graph will be compressed vertically.

To understand this, let's compare the graph of the regular sine function (y = sin(x)) with the given equation (y = (1/2)sin(2x)):

y = sin(x):

```
|
1 |° °
| ° °
0 |--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--
| 0 π 2π 3π 4π 5π 6π 7π 8π 9π 10π 11π 12π
```

y = (1/2)sin(2x):

```
|
0.5|° ° °
| ° ° ° ° ° ° ° °
0 |--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--
| 0 π 2π 3π 4π 5π 6π 7π 8π 9π 10π 11π 12π
```

As you can see, the graph of (1/2)sin(2x) is noticeably compressed vertically compared to the regular sine function. Therefore, we can conclude that the equation has a vertical compression by a factor of 1/2.

2. Horizontal compression by a factor of 2:
When we have a value greater than 1 as the coefficient inside the sine function, it results in a horizontal compression of the graph. In this case, the coefficient is 2, which means the graph will be compressed horizontally.

To understand this, let's compare the graph of the regular sine function (y = sin(x)) with the given equation (y = (1/2)sin(2x)):

y = sin(x):

```
|
1 |° °
| ° °
0 |--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--
| 0 π 2π 3π 4π 5π 6π 7π 8π 9π 10π 11π 12π
```

y = (1/2)sin(2x):

```
|
0.5|° ° ° ° ° ° °
| ° °
0 |--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--°--
| 0 π 2π 3π 4π 5π 6π 7π 8π 9π 10π 11π 12π
```

As you can see, the graph of (1/2)sin(2x) is compressed horizontally compared to the regular sine function. Therefore, we can conclude that the equation has a horizontal compression by a factor of 2.

In summary, the given equation "y = (1/2)sin(2x)" has a vertical compression by a factor of 1/2 and a horizontal compression by a factor of 2.