Question

1. dilation rules and properties

4a. which rule best represents a dilation with scale factor 1/2 about the origin?
a. (x,y)→(1/2x,1/2y)
b. (x,y)→(x + 1/2,y + 1/2)
c. (x,y)→(x,y - 2)
d. (x,y)→(-x,-y)
4b. which rule best represents an enlargement by scale factor 3?
a. (x,y)→(3x,3y)
b. (x,y)→(x - 3,y - 3)
c. (x,y)→(x,-y)
d. (x,y)→(1/2x,1/2y)
4c. which rule represents a reduction with scale factor 1/4?
a. (x,y)→(4x,4y)
b. (x,y)→(x + 1/4,y + 1/4)
c. (x,y)→(1/4x,1/4y)
d. (x,y)→(-x,y)
4d. which rule represents an enlargement with scale factor 5?
a. (x,y)→(x + 5,y + 5)
b. (x,y)→(5x,5y)
c. (x,y)→(x - 5,y - 5)
d. (x,y)→(-x,-y)

Explanation:

Step1: Recall dilation rules

For a dilation about the origin with scale - factor \(k\), the rule is \((x,y)\to(kx,ky)\). An enlargement has \(|k|> 1\) and a reduction has \(0 < |k|<1\).

Step2: Analyze 4a

A dilation about the origin with scale - factor \(\frac{1}{2}\) has the rule \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\). So the answer for 4a is A. \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\).

Step3: Analyze 4b

For a scale - factor of 3 (an enlargement), the rule is \((x,y)\to(3x,3y)\). So the answer for 4b is A. \((x,y)\to(3x,3y)\).

Step4: Analyze 4c

For a reduction with scale - factor \(\frac{1}{4}\), the rule is \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\). So the answer for 4c is A. \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\).

Step5: Analyze 4d

For an enlargement with scale - factor 5, the rule is \((x,y)\to(5x,5y)\). So the answer for 4d is B. \((x,y)\to(5x,5y)\).

Answer:

4a. A. \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)
4b. A. \((x,y)\to(3x,3y)\)
4c. A. \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\)
4d. B. \((x,y)\to(5x,5y)\)