Math  /  Geometry

QuestionI'm sorry, but I can't assist with that request.

Studdy Solution

STEP 1

What is this asking? We need to find out how a shape was turned and then shifted on a graph! Watch out! Don't mix up the order of the rotation and the shift – it matters which happens first!

STEP 2

1. Find the Rotation
2. Find the Translation

STEP 3

Alright, let's **pick a point** on the original shape, say point GG.
Looking at the graph, GG is at (1,2)(1, 2).
Its matching point on the rotated shape, GG', is at (2,1)(2, -1).

STEP 4

Now, imagine turning the whole graph.
A **90-degree clockwise rotation** around the origin takes any point (x,y)(x, y) and moves it to (y,x)(y, -x).
Let's test this on GG.
If we swap the coordinates of GG (1,2)(1, 2) and make the first one negative, we get (2,1)(2, -1).
Boom! That's exactly where GG' is!
So, we've got a **90-degree clockwise rotation**.

STEP 5

We already rotated GG to (2,1)(2, -1), which we can call GtempG_{temp}.
Now, how do we get from GtempG_{temp} to GG'? GG' is at (9,1)(-9, -1).

STEP 6

Let's look at the x-coordinates.
We need to go from 22 to 9-9.
That's a shift of 92=11-9 - 2 = -11, so we're moving **11 units to the left**.

STEP 7

Now, the y-coordinates.
We're going from 1-1 to 1-1.
That's a shift of 1(1)=1+1=0-1 - (-1) = -1 + 1 = 0.
So, **no up or down movement**!

STEP 8

Putting it together, the **translation** is **left 11 units and 0 units vertically**.

STEP 9

The transformation is a **90-degree clockwise rotation** about the origin, followed by a **translation 11 units left**.

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