Overview
This lecture reviews foundational algebra concepts for calculus, focusing on lines, their equations, slopes, angles of inclination, and the distance formula.
Lines and Slope
- A line is straight, has infinite points, and is uniquely determined by two points or one point and a slope.
- Slope (m) measures how a line rises or falls, calculated as: m = (y₂ - y₁) / (x₂ - x₁).
- Using two generic points (x₁, y₁) and (x₂, y₂), the slope formula applies to any line.
Point-Slope and Slope-Intercept Forms
- Fixing one point leads to the point-slope form: y - y₁ = m(x - x₁).
- Distributing and isolating y gives slope-intercept form: y = mx + b.
- In slope-intercept form, "m" is the slope and "b" is the y-intercept.
Special Lines
- y = constant describes a horizontal line (slope = 0).
- x = constant describes a vertical line (undefined slope).
- Standard form (Ax + By = C) can be rearranged to slope-intercept form for easier graphing.
Parallel and Perpendicular Lines
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals (m₁ = -1/m₂).
- Given a point and a required slope, use the point-slope form to find equations for lines parallel or perpendicular to a given line.
Angles of Inclination and Trigonometry
- The angle of inclination (θ) is the angle a line makes with the x-axis.
- Slope and angle are related: m = tan(θ).
- To find slope from angle: m = tan(θ); to find angle from slope: θ = arctan(m).
- Use the unit circle to find tangent values and corresponding angles.
Distance Formula
- Distance between two points: D = √[(x₂ - x₁)² + (y₂ - y₁)²].
- Derived from the Pythagorean theorem, using the horizontal and vertical differences as triangle legs.
Key Terms & Definitions
- Slope (m) — how steep a line is, calculated as change in y over change in x.
- Point-Slope Form — equation of a line: y - y₁ = m(x - x₁).
- Slope-Intercept Form — equation of a line: y = mx + b.
- Parallel Lines — lines with equal slopes.
- Perpendicular Lines — lines with slopes that are negative reciprocals.
- Angle of Inclination (θ) — angle a line makes with the x-axis.
- Distance Formula — finds length between two points: D = √[(x₂ - x₁)² + (y₂ - y₁)²].
Action Items / Next Steps
- Review and practice using the slope, point-slope, and distance formulas.
- Refresh knowledge of trigonometric functions (especially tangent and arctangent).
- Practice finding equations for parallel and perpendicular lines given a point and a slope.
- Memorize and use the unit circle for trigonometric calculations.