Introduction

• Welcome back to video number two

• Focus on deeper exploration of element lists and regression in relation to Desmos

• Shift from practical functions to the underlying math theory behind Desmos

• Importance for SAT problems emphasized

Understanding Regression in Desmos

• Regression Concept: Correlation of points through a statistical approach.

  • For effective regression, the number of data points should equal the number of constants.

• Illustration with Linear Regression:

  • Two points fit perfectly with two variables (M, B) in the equation format.

  • Example: With points (0, a) and (2, a + 3):

  • Using $y = mx + b$,

  • Define equations with these points.

    • 1. $1 = M * 0 + B$

    • 2. $3 = M * 2 + B$

  • Solving leads to: $B = 1$, $M = 1$ (after manipulating equations)

  • In Desmos, entering the same points would yield M = 1, B = 2.

• Key Insight: For accurate graphical representation, points must be perfectly aligned unless statistical regression is utilized.

Solving Systems of Equations in Desmos

• Example transformation of equations into Desmos format:

  • Direct transformation to show Desmos interpretation of equations.

• Notation Consistency:

  • Maintaining format: $y₁ = mx₁ + B$ during regression analysis ensures consistency.

• Importance of understanding that all expressions yield the same results, showcasing Desmos thinking.

General Behavior of Regression

• For fitting equations:

  • Same principle applies for quadratic and cubic functions.

  • Ideal: one point per variable, but accommodating extra points is possible if they align correctly.

  • Check fitting using R² statistics, where R² equals 1 implies perfect fit.

Application of Element Lists in Rewriting Expressions

• Example provided using an equation:

  • $ax + 3, 5x² - Bx + 4 = 20x³ - 9x² - 2x + 12$

• Shows property that outputs of left-hand expressions must match mapped output of the right-hand expressions for any value of x.

• Visual Representation: Graph alignment necessary for accurate fits on Desmos.

Utilizing Desmos for Active Regression and Graph Matching

• Simplifying complexities using point-wise regression setup.

• Example breakdown input:

  • Transition to using function format $f(x)$ to facilitate easier regression analysis.

  • Outcome visualization checks by plotting specific points and using Desmos functionalities.

• Quick Regression Solution: Show equation's direct correlation to point's transformations rather than prolonged calculation.

Practical Considerations for Functional Equivalence

• When comparing two expressions for equality, exploiting Desmos for visualization yields efficiency and accuracy in matching outputs.

  • Points extraction specified (1, 2, 3, etc.) create a wide range of checked fit points.

• Observation of RMSE (Root Mean Square Error): Tracking the error rate between actual and predicted values highlights graph performance accuracy.

Final Insights on Handling Complex Function Rewrites

• Goal: Rewrite into specified forms using Desmos effectively.

• Varied algebraic expressions can be visualized graphically for function verification.

• Challenges with Undefined Values: Recognizing limits to plotting facilitates understanding around mathematical domains and constraints.

• Prompts caution with equations leading to undefined outputs due to constraints on variables.

• Successful reliance on Desmos' regression reinforces understanding of function behavior and equivalence over defined ranges.

Conclusion

• Reinforces the application of regression techniques within Desmos and the importance of mathematical theory in preparing for standardized testing like the SAT.

• Encourages exploration of future Desmos videos to deepen understanding of these mathematical concepts and their applications.