unit 7: functions and transformations

Question 1

reflection over the y-axis (Ry)

Answer 1

(x,y) -> (-x,y)

Question 2

reflection over the x-axis (Rx)

Answer 2

(x,y) -> (x,-y)

Question 3

reflection over the line y=x (Ry=x)

Answer 3

(x,y) -> (y,x)

Question 4

reflection over the line y=-x (Ry=-x)

Answer 4

(x,y) -> (-y,-x)

Question 5

reflection over the origin or Rx*Ry

Answer 5

(x,y) -> (-x,-y)

Question 6

reflection over the line x=a (Rx=a)

Answer 6

(x,y) -> (2a-x,y)

Question 7

reflection over the line y=b (Ry=b)

Answer 7

(x,y) -> (x, 2b-y)

Question 8

dilate by 2 about the origin (D0,2)

Answer 8

(x,y) -> (2x,2y)

Question 9

dilate by -1/2 about the origin (D0,-1/2)

Answer 9

(x,y) -> (-1/2x,-1/2y)

Question 10

what does it mean when the dilation scale factor is negative or positive?

Answer 10

negative means that you do the dilation in the opposite direction of where the image is or opposite “side”

Question 11

translation by vector <a,b> (T<a,b>)

Answer 11

(x,y) -> (x+a,y+b)

Question 12

in translation, what is magnitude and how do you find it?

Answer 12

the magnitude is the length of the vector (or the distance travelled between the image and pre-image), you can find it by using Pythagorean theorem (a^2+b^2 ROOT)

Question 13

rotation by 90 degrees about the origin (R0,90) or (R0,-270)

Answer 13

(x,y) -> (-y,x)

Question 14

rotation by 270 degrees about the origin (R0,270) or (R0,-90)

Answer 14

(x,y) -> (y,-x)

Question 15

rotation by 180 degrees about the origin (R0,180) or (H0)

Answer 15

(x,y) -> (-x,-y)

Question 16

what is the difference between a negative rotation degree and a positive rotation degree?

Answer 16

a negative degree means a clockwise rotation, a positive degree means an anticlockwise rotation

Question 17

how do you solve a problem where the center of dilation or rotation is not the origin?

Answer 17

1. subtract the center from the image
2. implement the transformation as usual
3. add the center back

Question 18

label the transformation equation:
y=a*f(b(x+h))+k

Answer 18

-a=Rx, a>1=vertical stretch, 0<a<1=vertical compression
-b=Ry, horizontal dilation by the INVERSE of b (ie. b=2, dilate by 1/2)
-h=translate right, +h=translate left
-k= translate down, +k=translate up

Question 19

even vs. odd functions

Answer 19

even: f(x)=f(-x), symmetry over y axis
odd: f(-x)=-f(x), symmetry over origin

Question 20

(f-g)(x), (f*g)(x), (f+g)(x)

Answer 20

1. f(x)-g(x)
2. f(x) * g(x)
3. f(x) + g(x)

Question 21

how do you find the line of reflection?

Answer 21

connect the midpoints of all the images and pre images

Question 22

how do you find the center of rotation?

Answer 22

find the perpendicular bisectors of the images and pre images, where they intersect is the center of rotation

Question 23

how do you find the center of dilation?

Answer 23

connect the pre images and images, where these lines intersect is the center of dilation