Overview
This lecture covers how to find the equations of lines (linear equations), determine parallel and perpendicular lines, and apply these tools to real-world modeling using linear functions.
Slope-Intercept Form of a Line
- The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
- If given the slope (m) and the y-intercept (b), substitute into the formula to get the line's equation.
- If given a slope and a point, substitute the point into the formula to solve for b, then write the equation.
Finding the Equation from Points or Slope
- Given a slope and a point (x, y), use y = mx + b, plug in x and y, solve for b.
- Given two points, find the slope as (y2 - y1)/(x2 - x1), then use one point to solve for the equation.
- Alternatively, use the point-slope form: y - y₁ = m(x - x₁), then rearrange to slope-intercept form.
Parallel and Perpendicular Lines
- Non-vertical lines are parallel if they have the same slope and different y-intercepts.
- Two lines are perpendicular if the product of their slopes is -1; their slopes are opposite reciprocals.
- To check if lines are parallel or perpendicular, write each equation in y = mx + b form and compare slopes.
Finding Parallel or Perpendicular Lines Through a Point
- To find a parallel line, use the same slope as the original line and the given point.
- To find a perpendicular line, use the opposite reciprocal slope and the given point.
- Use the point-slope formula for both cases, then simplify.
Mathematical Modeling and Linear Models
- Mathematical modeling uses equations to predict outcomes based on data, commonly using linear models for simplicity.
- A scatter plot visually shows how well data fits a linear model.
- In modeling, an index is often used for years (e.g., x = 0 for 1980), simplifying calculations.
Example: Modeling GDP with a Linear Function
- Choose two data points; find the slope and write the linear equation using one point.
- The model equation predicts values for future years by substituting the appropriate x (years after base year).
- Always include correct units (e.g., trillions of dollars).
Key Terms & Definitions
- Linear Equation — An equation whose graph is a straight line.
- Slope (m) — The rate of change; rise over run.
- Y-intercept (b) — The y-value where the line crosses the y-axis.
- Point-Slope Form — The form y - y₁ = m(x - x₁) for a line through (x₁, y₁) with slope m.
- Parallel Lines — Lines with the same slope, different y-intercepts.
- Perpendicular Lines — Lines with slopes that are opposite reciprocals (m₁ × m₂ = -1).
- Mathematical Model — An equation used to represent and predict real-world behavior.
- Index — A variable representing years after a base year, used in modeling.
Action Items / Next Steps
- Practice finding equations of lines given various data (slope, points, etc.).
- Try exercises on determining if lines are parallel or perpendicular.
- Model real-world data with linear equations using scatter plots and indexes.
- Review definitions and ensure you can move between different forms of linear equations.