Transforming functions

Learning objectives Translate functions left and right left = f(x+…), right = f(x-…) Translate functions up and down up = f(x)+…, down = f(x)-… Stretch functions parallel to y-axis 2f(x), ½f(x) etc. Stretch functions parallel to x-axis f(2x), f(½x) etc.

f(x) f(x) = x2 – 4 x f(-2) = (-2)2 – 4 = 0 f(2) = 22 – 4 = 0 f(-1) = (-1)2 – 4 = -3 f(1) = (1)2 – 4 = -3 f(0) = 02 – 4 = -4

Drawing 2f(x) … f(x) f(x) = x2 – 4 x 2f(-2) = 20 = 0 Stretch x2 away from the x-axis (Double y coordinates) 2f(0) = 2(-4) = -8

Drawing ½f(x) … f(x) f(x) = x2 – 4 x ½f(-2) = ½0 = 0 Squash by ½ towards the x-axis (Halve y coordinates)

( ) Drawing f(x)+3 … f(2)+3 = 0+3 = 3 f(x) f(-2)+3 = 0+3 = 3 x f(x) = x2 – 4 ( ) Translate 0 3

( ) Drawing f(x+3) … f(-1+3) = f(2) = 0 f(x) f(-5+3) = f(-2) = 0 f(x) = x2 – 4 x f(-4+3) = f(-1) = -3 f(-2+3) = f(1) = -3 f(-3+3) = f(0) = -4 ( ) Translate -3

( ) Drawing f(x)-1 … f(x) f(x) = x2 – 4 x f(-2)-1 = 0-1 = -1 ( ) Translate 0 -1

( ) Drawing f(x-1) … f(x) f(3-1) = f(2) = 0 f(-1-1) = f(-2) = 0 x f(x) = x2 – 4 f(1-1) = f(0) = -4 ( ) Translate 1

Drawing f(2x) … f(x) f(x) = x2 – 4 x Squash by ½ towards the y-axis (Halve x coordinates)

Drawing f(½x) … f(x) f(x) = x2 – 4 x Stretch x2 away from the y-axis (Double x coordinates)

Recap learning objectives Translate functions left and right left = f(x+…), right = f(x-…) Translate functions up and down up = f(x)+…, down = f(x)-…

Recap learning objectives Stretch functions parallel to y-axis 2f(x), ½f(x) etc. Stretch functions parallel to x-axis f(2x), f(½x) etc. And finally …

f(-x) … Reflection in y-axis Drawing f(-x) … f(x) f(-x) … Reflection in y-axis

-f(x) … Reflection in x-axis Drawing -f(x) … f(x) -f(x) … Reflection in x-axis

-f(-x) … Rotation 180° about (0,0) Drawing -f(-x) … f(x) -f(-x) … Rotation 180° about (0,0)