Question

01/07 - rotations & dilation practice l
possible points: 5.05
the pre - image (3, -1) dilated from the origin using the rule ((x, y) \to (2x, 2y)) has an image of ((square), (square)), and scale factor of (square)
the pre - image (6, 0) dilated from the origin with a scale factor of 1/3 has an image of ((square), (square)). the rule used for this transformation is ((x, y) \to (square x, square y))
the pre - image ((square), (square)) dilated from the origin using the rule ((x, y) \to (\frac{1}{4}x, \frac{1}{4}y)) has an image of (8, 4). the scale factor for this dilation is (square)

Explanation:

First Problem:

Step1: Apply dilation rule to (3, -1)

Using the rule \((x, y) \to (2x, 2y)\), substitute \(x = 3\) and \(y=-1\). So \(2x = 2\times3 = 6\) and \(2y = 2\times(-1)=-2\). The scale factor is the multiplier, which is 2.

Step2: Determine image and scale factor

The image is \((6, -2)\) and scale factor is 2.

Second Problem:

Step1: Dilate (6, 0) with scale factor \( \frac{1}{3} \)

For a dilation with scale factor \(k\), the rule is \((x, y) \to (kx, ky)\). Here \(k=\frac{1}{3}\), \(x = 6\), \(y = 0\). So \(kx=\frac{1}{3}\times6 = 2\) and \(ky=\frac{1}{3}\times0 = 0\). The rule is \((x, y) \to (\frac{1}{3}x, \frac{1}{3}y)\).

Step2: Find image and rule

The image is \((2, 0)\) and the rule has \(k=\frac{1}{3}\) for \(x\) and \(y\).

Third Problem:

Step1: Find pre - image from rule \((x, y) \to (\frac{1}{4}x, \frac{1}{4}y)\) with image (8, 4)

Let the pre - image be \((x, y)\). We know that \(\frac{1}{4}x=8\) and \(\frac{1}{4}y = 4\). Solve for \(x\): \(x=8\times4 = 32\). Solve for \(y\): \(y = 4\times4=16\). The scale factor is \(\frac{1}{4}\) (from the rule \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\)).

Step2: Determine pre - image and scale factor

The pre - image is \((32, 16)\) and the scale factor is \(\frac{1}{4}\).

Answer:

1. Image: \(\boldsymbol{(6, -2)}\), Scale Factor: \(\boldsymbol{2}\)
2. Image: \(\boldsymbol{(2, 0)}\), Rule: \((x, y)\to(\boldsymbol{\frac{1}{3}}x, \boldsymbol{\frac{1}{3}}y)\)
3. Pre - image: \(\boldsymbol{(32, 16)}\), Scale Factor: \(\boldsymbol{\frac{1}{4}}\)