1. PreCalculus Transformed and Nspired OCTM, October 28, 2016
Handouts posted at
Chris Harrow, Hawken School or
@chris_harrow

2. Variations on traditional transformations …
Name a function through (2,1) & (6,81)
Line y=a+bx ?
Standard approach is find slope & either use point- slope or plug and solve for b.
High level transformations: Lines add!
Alternatives:
System of equations (note “free” definition of slope)
I really wish I had a y-intercept … I think I’ll “make” one.
Are there any other options with 2 parameters?

3. Variations on traditional transformations …
Name a function through (2,1) & (6,81)
Lines are easy … What about other functions?
Power y = a*x^b
Systems
Transformations
Exponential y=a*b^x
High level transformations: Exponentials multiply!
Lots more … anything with 2 parameters

4. Extending Transformations
We all know constant stretches and slides.
Transformations change pre-images.
Functions change inputs/pre-images.
Transformations  Functions
Now what???

5. Trig Bouncing I What do a, b, & k control?
Don’t think of sinusoids as images of fixed transformation. Imagine a dynamic curve: oscillating and bouncing within given bounds.
Ceilings and Floors
Midlines (sinusoidal axes)
Amplitude versus “Ceiling Vector”

6. Trig Bouncing II No more need for reflections
The addend is the “x-axis”
The coefficient is the “amplitude”

7. Trig Bouncing III Apply ceilings & floors to polar functions
No longer a need to memorize various limacons, cardioids, & rose curves.
Hybrid graphs.
Nspire file: Intro Polar

8. The Reciprocal Transformation (REC)
What happens when you reciprocate a graph?
That is, given a graph of , what is ?
Any fixed points?
Any destructive points?
Small vs. big?
Positive vs. negative?

9. Polynomials  Rationals
There are 3 basic ways polynomials touch the x-axis:
Linear
Bounce
“Wiggle”
Or is that just two? Odd & Even
BIG INSIGHT: There are two types of VAs

10. Local vs. End behavior

11. REC(Trig)
Graph
Why do all parent trig graphs have odd VAs?

12. Translate or re-center?
Many teach this with a temporary x-axis.
The function behaves the same with respect to x-axis, no matter where it is located.
Consider transforming the x-axis.

13. Variable EBA
Form:
There’s only one EBA: To teach HA, slant, etc. is to overcomplicate
What if q(x) isn’t constant? Bend the Asymptote!

14. REC(Exponentials)
Graph

15. Logistic functions
Graph