Overview
This lecture covers essential concepts and problem-solving techniques related to linear functions, including slope calculations, graphing, and writing equations in various forms.
Calculating Slope Between Two Points
- The slope formula between points ((x_1, y_1)) and ((x_2, y_2)) is ((y_2 - y_1) / (x_2 - x_1)).
- Example: Slope through (2, -3) and (4, 5) is (4).
Slope and Y-Intercept from an Equation
- Slope-intercept form is (y = mx + b), where (m) is slope and (b) is y-intercept.
- In (y = 2x - 3), the slope is 2, and the y-intercept is -3 (or (0, -3)).
Graphing Special Linear Equations
- (x = 2) graphs as a vertical line at (x = 2).
- (y = 3) graphs as a horizontal line at (y = 3).
Graphing with Slope-Intercept Method
- In (y = 3x - 2), slope is 3 and y-intercept is (0, -2).
- Start at (0, -2); for each unit right, move up 3 units, plot additional points, and connect with a line.
Graphing from X- and Y-Intercepts (Standard Form)
- For (2x - 3y = 6): x-intercept is (3, 0), y-intercept is (0, -2).
- Plot both points and draw a line through them.
Writing Equations with Point-Slope or Slope-Intercept Form
- Point-slope form: (y - y_1 = m(x - x_1)).
- Example: Through (2, 5) with slope 3: (y - 5 = 3(x - 2)). Slope-intercept: (y = 3x - 1).
Writing the Equation through Two Points
- Find the slope first using the two points.
- For (−3, 1) and (2, −4): Slope is −1; equation in point-slope is (y - 1 = -1(x + 3)); in slope-intercept: (y = -x - 2).
Equation of a Line Parallel to a Given Line
- Parallel lines have the same slope.
- Convert the given equation to slope-intercept to find the slope, then use point-slope form with the provided point.
Equation of a Line Perpendicular to a Given Line
- Perpendicular lines have slopes that are negative reciprocals.
- Given line’s slope is 3/4; perpendicular slope is −4/3.
- Use point-slope form with the given point and perpendicular slope; convert to slope-intercept if needed.
Key Terms & Definitions
- Slope ((m)) — the steepness of a line, calculated as rise over run.
- Y-intercept ((b)) — point where the line crosses the y-axis.
- Slope-intercept form — (y = mx + b).
- Standard form — (Ax + By = C).
- Point-slope form — (y - y_1 = m(x - x_1)).
- Parallel lines — lines with equal slopes.
- Perpendicular lines — slopes are negative reciprocals.
Action Items / Next Steps
- Practice identifying slope and intercepts from linear equations.
- Graph linear equations using all three major forms.
- Complete assigned homework problems on writing and graphing linear equations.