1. Quadratic Explorations & CAS
USACAS July, 2017
Chris Harrow
Hawken School
@chris_harrow

2. Given three points, determine an equation of a quadratic.
By hand, choose your form wisely.
The power of CAS makes form irrelevant. You can directly calculate any aspect you want!
What about horizontal parabolas?
QUADRATICS & CAS

3. But who said the parabola had to be horizontal or vertical?
With CAS to handle the heavy algebraic transformations, you can create AND ANIMATE all parabolas through three given points.
A full explanation of this derivation and a video is available on my ‘blog:
QUADRATICS & CAS
Bonus Content:
Rotations
Download my GeoGebra file that you can animate to show the rotating parabolas

4. Math is the “Science of Patterns”
Exploring functions
Little fact boxes litter most textbooks; why not engage higher level thinking with pattern discovery?
DON’T TELL … DO AN EXPERIMENT!
Desmos and Nspire are limited.
GeoGebra creates points easily and traces points AND curves
THE POWER OF SLIDERS
Math is the “Science of Patterns”

5. EXPLORING BASIC QUADRATICS
Thanks to an early USACAS presenter.
Simplify initial exploration by setting a=1, b=0, & c=0.
What happens when you vary c?
What happens when you vary a?
What happens when you vary b?
Proof via CAS
EXPLORING BASIC QUADRATICS
𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄
Upload my
GeoGebra file or
easily create
your own!

6. EXPLORING BASIC QUADRATICS, 2
At least we understand quadratics now … or do we? (Thanks to Doug Kuhlmann)
Make b non-zero. Now what happens when you vary a?
Now trace the parabola instead of the vertex!
The power of degenerate forms.
EXPLORING BASIC QUADRATICS, 2
We look for patterns we already recognize!
Wake up!!!
𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄

7. AN INNOCENT REFRESHING OF A PREDICTABLE PROBLEM
Systems of equations
Summer School Algebra 2 question:
Create a system of two quadratic functions whose only solution is the point (1,1).
AN INNOCENT REFRESHING OF A PREDICTABLE PROBLEM

8. Slightly overdone, but what a nice pattern! Does it generalize?
System of equations
Slightly overdone, but what a nice pattern! Does it generalize?
PROOF?
WHAT A STUDENT THOUGHT
𝑦= 𝑥 2 +𝑥−1 𝑦= 𝑥 2 𝑦= 𝑥 2 +2𝑥−2

9. Thanks to Michael Buescher Leverage x=1 symmetry
WHAT A TEACHER THOUGHT
𝑦= 𝑥 2 +𝑎 𝑥−1 𝑦= 𝑥−2 2 +𝑎 𝑥−1

10. WHAT ANOTHER TEACHER SUGGESTED
Inspired by Steve Earth
Consider all parabolas that pass through (1,1)
So a family of quadratics all containing (1,1) is:
Explore on Grapher with Sliders
WHAT ANOTHER TEACHER SUGGESTED
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐| 𝑥,𝑦 = 1,1 1=𝑎+𝑏+𝑐 𝑐=1−𝑎−𝑏
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐|𝑐=1−𝑎−𝑏 𝑦=𝑎 𝑥 2 +𝑏𝑥+ 1−𝑎−𝑏

11. WHAT ANOTHER TEACHER SUGGESTED, 2
Explore on Grapher with Sliders
Vary b … we’ve seen that
Vary a … What’s happening? Can you prove it?
WHAT ANOTHER TEACHER SUGGESTED, 2

12. GRAVITY & QUADRATICS not CAS, but cool!
Use a probe to gather height of a bouncing ball.
Then slice out first bounce (leave out the endpoints..Why?)
GRAVITY & QUADRATICS not CAS, but cool!
Bounce data available online
Formats
- Excel
- TI-Nspire CAS

13. What are the units and what does it look like?
Now compute the slopes between each successive pair of points. (This is a new y assigned to the x-value of each pair.)
What are the units and what does it look like?
GRAVITY & CAS, 2

14. Units are (sec, m/sec). This is a graph of velocity.
It’s way too pretty.
Compute a linear regression.
GRAVITY & CAS, 3

15. GRAVITY & CAS, 4