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QUICK REVIEW OF ROTATION EXAMPLES

Remember, to rotate a shape around its own point, you need to pick a point on the shape, and then rotate it.

Sounds pretty straight forward right?

Here’s the example from yesterday

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ROTATING A SHAPE FROM ITS OWN POINT

So let’s take a look at this triangle that we have here.

We have a point at the tip of the triangle (4, 4) and we would like to rotate the triangle from that point.

As we can see, this point is on the triangle, so it should be no problem right?

Let’s start off by rotating the triangle by 90 degrees.

So as you can see, the triangle has been rotated by 90 degrees by that same point.

So how did we do it?

Think of it like sticking a pin on the point, and rotating it 90 degrees.

Like this!

Now let’s try rotating that same triangle another 90 degrees.

Again, think about putting a pin in the point and spinning it 90 degrees again. �So, as we can see the blue triangle is 90 degrees from the red triangle, and the green triangle is 180 degrees from the red triangle.

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ROTATING ABOUT A POINT AWAY FROM THE OBJECT

Again, you can always rotate the object from a point away from the object.

Here’s the example from yesterday:

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Other types of rotation

You can also rotate an object by a point away from the object, however this type of rotation is a little more advanced.

Here is an example of this type or rotation.

Let’s say we have a triangle, and a point away from that triangle.

Now, let’s say we want to rotate that triangle around that point by say, 45 degrees?

Here’s what it would look like:

Now, let’s keep it going by rotating 45 degrees until...

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Important note about rotations

If you rotate an object around the origin by:

90 degrees: You flip the coordinates.

Ex:

Let’s say we have a triangle

We can see the triangles points are:

(1,0), (4,0), (4,3)

And we rotated that triangle 90^o about the origin

We can see the point coordinates are flipped!

The new points for the new triangle are:�(0,1), (0,4), (-3,4)�So why did 3 turn to -3?

We’re in quadrant II!

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So, what about other rotations?

Basically, if you rotate an object by a multiple of 90, you will keep switching the coordinates.

So, for 180, you’ll switch the coordinates twice,

Like this:

Now we can see that our new points went from:

(1,0), (4,0), (4,3)

(-1,0), (-4,0), (-4,-3)

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So what about reflections?

Reflections are very similar to rotations, but a little bit different.

A reflection is exactly what you would think of when you look in a mirror, it’s a projected image of a shape that is symmetrical to its origin about the plane of reflection.

So what that means in plain English, is basically a line that causes the exact same , but mirrored, object to appear on each side of it.

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See, now that is a perfect example of a reflection.

The mirror acts as the line of reflection, and the mirror image is the actual reflection.

This, however, is not a reflection. The image is a mirror image, however it doesn’t stay that way…..

(I know she’s supposed to be good and all, but that’s still creepy.)

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YOU CAN ALSO CHANGE HOW REFLECTIONS WORK

It’s all about the line of reflection (it’s actually called a plane of reflection usually, but we’ll call it a line for right now).

For example, let’s say we have this figure:

And we toss in a line

And then, for fun, we reflect about that line.

Seems pretty easy right?

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Reflection inside the object

But what if, say, we decide to reflect the object inside of itself?

So again, we have an object

But this time, the line is actually through the object

Now when we reflect about this line, it’s going to get a little weird.

See? I told you!�However, if you notice, the green star is a true mirror image of the blue star through the line of reflection.

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Coordinate Rules

So, reflecting across a line is great and all, but what if we don’t have a graph?

What if we just have some coordinates, and need to figure out what’s happening?

This is where the coordinate rules come into play (just like last time).

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Reflecting about the x-axis:

So, when we are reflecting about the x-axis, what coordinates are actually changing?

Does the x-coordinate change?

Does the y-coordinate change?

Does neither?

Well let’s look:

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Example:

Reflect this object about the x-axis:

First, we need to know, what is the line of reflection?

Where can we find it?

Well, we know it’s the x-axis right?

So:

So it should look something like this:

As we can see, the shape seems to flip over the x-axis.

So let’s take a look at the coordinates:

A (1, 1) but A’ (1, -1)

B (3, 1) but B’ (3, -1)

C (3, 3) but C’ (3, -3)

So what changed?

We just multiplied the y-coordinate by -1.

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AND THAT’S THE RULE

When reflecting about the x-axis, we multiply the y-coordinate of each point by -1.

So what do you think we’ll do if we reflect about the y-axis?

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EXAMPLE: