Overview
This lecture reviews linear functions, focusing on forms of their equations, the concept of slope, and how to find equations for lines through given points.
Linear Functions & Slope
- Linear functions are written as ( f(x) = mx + b ) or ( y = mx + b ).
- The slope ( m ) is defined as the change in ( y ) over the change in ( x ) (( \Delta y / \Delta x )).
- Slope formula: ( m = (y_2 - y_1) / (x_2 - x_1) ).
- Slope for linear functions is constant, unlike quadratic or square root functions.
- "Rise over run" describes slope: the amount ( y ) increases per unit increase in ( x ).
Special Types of Lines
- Horizontal lines: ( y = b ), slope (( m )) is 0.
- Vertical lines: ( x = c ), slope is undefined and not considered a function.
Slope Examples
- To find slope between points (3, 5) and (-7, 2): ( (2-5)/(-7-3) = (-3)/(-10) = 3/10 ).
- Positive slope: moves up to the right; negative slope: moves down to the right.
Forms of Linear Equations
- Slope-intercept form: ( y = mx + b ), where ( m ) is slope, ( b ) is the ( y )-intercept.
- Point-slope form: ( y - y_1 = m(x - x_1) ), for a line through point ( (x_1, y_1) ).
- General form: ( Ax + By = C ), commonly used for systems of equations.
Example: Finding a Line's Equation
- Given points (2, 5) and (1, -2), find slope: ( (-2-5)/(1-2) = -7/(-1) = 7 ).
- Point-slope form: ( y - 5 = 7(x - 2) ).
- Slope-intercept form: ( y = 7x - 9 ).
- General form: ( -7x + y = -9 ).
Key Terms & Definitions
- Linear Function — A function of the form ( f(x) = mx + b ) whose graph is a straight line.
- Slope (( m )) — The ratio of the change in ( y ) to the change in ( x ) between two points.
- Slope-intercept Form — ( y = mx + b ), expresses slope and ( y )-intercept.
- Point-slope Form — ( y - y_1 = m(x - x_1) ), uses a point and slope.
- General Form — ( Ax + By = C ), linear equation with constants.
Action Items / Next Steps
- Review the forms of linear equations and practice converting between them.
- Try finding the equation of a line given two points.
- Prepare any questions for clarification in the next session.