unit 7: functions and transformations Flashcards
reflection over the y-axis (Ry)
(x,y) -> (-x,y)
reflection over the x-axis (Rx)
(x,y) -> (x,-y)
reflection over the line y=x (Ry=x)
(x,y) -> (y,x)
reflection over the line y=-x (Ry=-x)
(x,y) -> (-y,-x)
reflection over the origin or Rx*Ry
(x,y) -> (-x,-y)
reflection over the line x=a (Rx=a)
(x,y) -> (2a-x,y)
reflection over the line y=b (Ry=b)
(x,y) -> (x, 2b-y)
dilate by 2 about the origin (D0,2)
(x,y) -> (2x,2y)
dilate by -1/2 about the origin (D0,-1/2)
(x,y) -> (-1/2x,-1/2y)
what does it mean when the dilation scale factor is negative or positive?
negative means that you do the dilation in the opposite direction of where the image is or opposite “side”
translation by vector <a,b> (T<a,b>)
(x,y) -> (x+a,y+b)
in translation, what is magnitude and how do you find it?
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)
rotation by 90 degrees about the origin (R0,90) or (R0,-270)
(x,y) -> (-y,x)
rotation by 270 degrees about the origin (R0,270) or (R0,-90)
(x,y) -> (y,-x)
rotation by 180 degrees about the origin (R0,180) or (H0)
(x,y) -> (-x,-y)
what is the difference between a negative rotation degree and a positive rotation degree?
a negative degree means a clockwise rotation, a positive degree means an anticlockwise rotation
how do you solve a problem where the center of dilation or rotation is not the origin?
- subtract the center from the image
- implement the transformation as usual
- add the center back
label the transformation equation:
y=a*f(b(x+h))+k
-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
even vs. odd functions
even: f(x)=f(-x), symmetry over y axis
odd: f(-x)=-f(x), symmetry over origin
(f-g)(x), (f*g)(x), (f+g)(x)
- f(x)-g(x)
- f(x) * g(x)
- f(x) + g(x)
how do you find the line of reflection?
connect the midpoints of all the images and pre images
how do you find the center of rotation?
find the perpendicular bisectors of the images and pre images, where they intersect is the center of rotation
how do you find the center of dilation?
connect the pre images and images, where these lines intersect is the center of dilation