Which of the following transformations will result in an image that maps onto itself? a. rotate 360 degrees. b. reflect across the y-axis, and then reflect across the x-axis. c. rotate 270 degrees counterclockwise, and the reflect it across the x-axis. d. rotate 90 degrees counterclockwise, and then rotate 180 degrees.

A. Rotate 360°

Step-by-step explanation:

We are required to find the transformation mapping an image onto itself.

According to the options, we have,

A. Rotate 360°.

Rotating 360° changes (x,y) to (x,y).

That is, this transformation will map the image onto itself.

B. Reflect across the y-axis, and then reflect across the x-axis.

Reflecting (x,y) across y-axis gives the point (-x,y).

Reflecting (-x,y) across x-axis gives the point (-x,-y).

So, this transformation maps (x,y) to (-x,-y).

C. Rotate 270° counterclockwise, and then reflect it across the x-axis.

Rotating 270° counter-clockwise changes (x,y) to (y,-x).

Reflecting (y,-x) across x-axis gives the point (y,x).

So, this transformation maps (x,y) to (y,x).

D. Rotate 90° counterclockwise, and then rotate 180°.

Rotating 90° counter-clockwise changes (x,y) to (-y,x).

Rotating (-y,x) by 180° counter-clockwise gives (y,-x).

So, this transformation maps (x,y) to (y,-x).

Hence, transformation in option A maps the image onto itself.

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

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step-by-step explanation: