1. Function Transformations
Lecture: 2C Pre AP & GT Precalculus

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

3. Homework Review Function is decreasing without concavityX 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):

4. Homework Review Function is increasing without concavityX 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):

5. Homework Review Function is Increasing & Concave UpX 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

6. Vertical Shift
Rigid Transformation

7. Horizontal Shift
Rigid Transformation

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

9. Horizontal Shrinks & Stretches
Non-Rigid Transformation

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

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

12. 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↓

13. 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↓

14. 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↓

15. 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)

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