Fitting Quadratic Functions to Data and Intercept Form

This lesson focuses on the intercept form of quadratic equations and how to fit quadratic functions to experimental data. The key concepts covered include:

• Understanding x-intercepts and their role in the intercept form
• The general equation for the intercept form of a quadratic function
• Practical examples of fitting quadratic models to lab data
• Solving for the 'a' coefficient in the intercept form equation
• Converting between different forms of quadratic equations

Definition: The intercept form of a quadratic function is f(x) = a(x-x₁)(x-x₂), where x₁ and x₂ are the x-intercepts of the function.

Highlight: X-intercepts are the points where the graph of a quadratic function crosses the x-axis.

The lesson provides two detailed examples:

1. Fitting a quadratic model to scientific lab data
2. Writing a quadratic function from a given graph

Both examples demonstrate the process of identifying x-intercepts, using a given point to solve for the 'a' coefficient, and expressing the final function in intercept form.

Example: For the lab data, the function in intercept form was determined to be f(x) = -21/8(x-12)(x-3).

Vocabulary:

• Vertex form: f(x) = a(x-h)² + k
• Standard form: f(x) = ax² + bx + c

The lesson emphasizes the importance of understanding how to convert between different forms of quadratic equations and how to apply these concepts to real-world data analysis scenarios.

Highlight: The process of fitting a quadratic function to data involves identifying key points (such as x-intercepts and a vertex) and using these to determine the coefficients of the quadratic equation.

This material provides a solid foundation for students learning about quadratic regression and curve fitting, which are essential skills in data analysis and scientific modeling.