2.4 Transformations of Functions

Fill out the chart as we go along… Transformation f(x)

Vertical Shift y = f(x) + c or y = f(x) – c up ‘c’ units down ‘c’ units EX: y = x2 and y = x2 - 2 F(x) x y -2 4 -1 1 2 F(x)-2 x y -2 -1 1 2

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c

Horizontal Shift EX g(x) = (x + 4)2 y = f(x + c) or y = f(x – c) left ‘c’ units right ‘c’ units EX g(x) = (x + 4)2 f (x) x y -2 4 -1 1 2 f (x+4) x y

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c)

Graph: y = (x – 2)2 + 3 f (x) x y -2 4 -1 1 2 f (x-2)+3 x y

Reflecting Graphs y = f(x) or y = -f(x) The y-coordinate of each point of the graph of y = -f(x) is the negative of the y- coordinate of the corresponding on y = f(x). Reflection in the x-axis. f (x) x y -2 4 -1 1 2 -f (x) x y -2 -1 1 2

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x)

y = f(-x) Reflection in the y-axis -4 DNE -1 1 4 2 x y

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x) Reflection across y-axis f(-x)

Vertical Stretch & Shrink y= cf(x) (y-coordinate multiplied by ‘c’) If c > 1 – stretch by a factor of ‘c’ If 0 < c < 1 – shrink vertically by a factor of ‘c’

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x) Reflection across y-axis f(-x) Vertical stretch cf(x) if c>1, stretch if 0<c<1, shrink

EX x y -2 4 -1 1 2 x y x y -2 -1 1 2

EX x y -4 DNE -1 1 4 2 x y

Horizontal Stretch & Shrink y = f(cx) (x-coordinate multiplied by ‘c’)

Fill out the chart as we go along… Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x) Reflection across y-axis f(-x) Vertical stretch cf(x) if c>1, stretch if 0<c<1, shrink Horizontal Stretch f(cx) if c>1, shrink if 0<c<1, stretch

Ex Given: find y = f(2x) and y = f(1/2x)

EX x y -2 4 -1 1 2 x y x y

Even and Odd Functions Even if f(-x) = f(x) Odd if f(-x) = -f(x) Symmetric with respect to y-axis Odd if f(-x) = -f(x) Symmetric with respect to the origin (rotate 180º about the origin or reflect 1st in x-axis and then in y-axis.)

Ex even/odd/neither f(x) = x3 + x f(x) = 7 – x6 f(x) = 3x – x3

Make sure your chart is complete!! Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x) Reflection across y-axis f(-x) Vertical stretch cf(x) if c>1, stretch if 0<c<1, shrink Horizontal Stretch f(cx) if c>1, shrink if 0<c<1, stretch

Homework Pg 191 #1-35 odd, 41, 43, 61-66