Function Transformations Lecture: 2C Pre AP & GT Precalculus

Agenda Homework Review Rigid Transformations Shifts (Vertical & Horizontal) Non-Rigid Transformations Shrinks & Stretches What does it mean?

Homework Review Function is decreasing without concavity X 1 2 3 4 5 Y −3 −7 −11 −15 1st Diff −4 2nd Diff Function is decreasing without concavity Function is linear and has algebraic form (not required):

Homework Review Function is increasing without concavity X 1 2 3 4 5 Y 10 16 22 28 34 40 1st Diff 6 2nd Diff Function is increasing without concavity Function is linear and has algebraic form (not required):

Homework Review Function is Increasing & Concave Up X 1 2 3 4 5 y 6 12 20 30 42 56 1st Diff +6 +8 +10 +12 +14 2nd Diff +2 Function is Increasing & Concave Up Because 2nd Diff is Constant, Function is Quadratic

Vertical Shift Rigid Transformation

Horizontal Shift Rigid Transformation

Flips Rigid Transformation About the Y-axis: About the X-axis:

Horizontal Shrinks & Stretches Non-Rigid Transformation

Vertical Shrinks & Stretches Non-Rigid Transformation c f(x) If c>1 stretch If c<1 shrink

Examples 13 units to the left: 2 to the right and 8 down: _ 10 up, reflected about x-axis:

Example Let’s use origin as reference for shifts (1,0) →(2,0) No, Vertical Shift No, Horizontal Shift (1,0) →(2,0) With same y-values what do we do to get from new to old From 2 to 1 multiply by ½ So HStretch of one-half ↑New Original↓

Example Let’s use origin as reference for shifts Flip No, Vertical Shift No, Horizontal Shift Flip Yes, about y VStretch: (1,1) →(-4,1) Stretch: (1,1) →(2,1) New to Old so ½ ↑New Original↓

Example Parent Function: Hshift: NO Vshift: Yes, Down 2 VStretch: Absolute Value Hshift: NO Vshift: Yes, Down 2 VStretch: (2,2) →(2,-1)→(2,1), +2 Flip: Yes, x ↑New Original↓

What does it mean? I wanted to paint my house and bought F(x) gallons Which transformation corresponds to: I bought two extra gallons just in case: F(x) +2 I bought enough paint to cover my house twice: 2 F(x) I bought enough paint for my house plus my mailbox which is 2 square feet: F(x+2)

Homework Pg 80 V6, #13-18 (A) Pg 81 #44-50 (even) Pg 83 #69,70,73