Studdy AI Logo
Math  /  Geometry

QuestionGraph the triangle with vertices X(−3,2),Y(2,3)X(-3,2), Y(2,3), and Z(1,−1)Z(1,-1) and its image after a dilation with scale factor k=−3k=-3.

Studdy Solution

STEP 1

What is this asking? We need to draw a triangle and then make it **three times bigger**, but also **flip it** to the other side of the origin because the scale factor is negative! Watch out! Don't forget that a negative scale factor means the image will be on the opposite side of the origin compared to the original triangle.

STEP 2

1. Plot the original triangle
2. Dilate the vertices
3. Plot the image triangle

STEP 3

Let's **plot** the original triangle XYZ.
We have the coordinates X(−3,2)X(-3,2), Y(2,3)Y(2,3), and Z(1,−1)Z(1,-1).
Imagine a graph, and put those points on it!
Connect the dots, and bam!
You've got triangle XYZ.

STEP 4

Now for the **dilation**!
Our **scale factor** is k=−3k = -3.
Remember, this means we're multiplying each coordinate of our original points by −3-3.

STEP 5

Let's start with point X(−3,2)(-3, 2).
Multiply each coordinate by −3-3: (−3⋅−3,2⋅−3)=(9,−6)(-3 \cdot -3, 2 \cdot -3) = (9, -6) So, our new point X', which is X after dilation, is at (9,−6)(9, -6).

STEP 6

Next up is Y(2,3)(2, 3).
Multiply each coordinate by −3-3: (2⋅−3,3⋅−3)=(−6,−9)(2 \cdot -3, 3 \cdot -3) = (-6, -9) Our new point Y' is at (−6,−9)(-6, -9).

STEP 7

Finally, let's dilate Z(1,−1)(1, -1).
Multiply each coordinate by −3-3: (1⋅−3,−1⋅−3)=(−3,3)(1 \cdot -3, -1 \cdot -3) = (-3, 3) Our new point Z' is at (−3,3)(-3, 3).

STEP 8

Alright, we've got our **dilated points**: X'(9,−6)(9, -6), Y'(−6,−9)(-6, -9), and Z'(−3,3)(-3, 3).
Now, **plot** these new points on the same graph as the original triangle.

STEP 9

Connect X' to Y', Y' to Z', and Z' back to X'.
Boom! You've just drawn the **image** of triangle XYZ after a dilation with a scale factor of −3-3.
Notice how it's **three times bigger** than the original and **flipped** across the origin!

STEP 10

The image of triangle XYZ after a dilation with a scale factor of −3-3 has vertices at X'(9,−6)(9, -6), Y'(−6,−9)(-6, -9), and Z'(−3,3)(-3, 3).
Plot the original points and connect them, then plot the new points and connect them, and you'll see the magic of dilation!

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord