PreCalculus Transformed and Nspired OCTM, October 28, 2016

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Presentation transcript:

PreCalculus Transformed and Nspired OCTM, October 28, 2016 Handouts posted at https://casmusings.wordpress.com/ Chris Harrow, Hawken School cdharr@hawken.edu or casmusings@gmail.com @chris_harrow

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?

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

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

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”

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

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

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?

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

Local vs. End behavior

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

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.

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!

REC(Exponentials) Graph

Logistic functions Graph