Linear Functions Lecture Notes

Course Logistics & Mini Exam Overview

• Schedule: Section 1.4 completed; Section 1.5 covered Wednesday.

• Mini Exam 1: Covers Sections 1.1 through 1.5 on Friday.

  • Begins with $20$ to $25$ minutes of review.

  • Format: Closed-note and no-calculator.

Forms of Linear Functions

• Point-Slope Form: $y - y_0 = m(x - x_0)$

  • Efficient when given slope $m$ and point $(x_0, y_0)$.

• Slope-Intercept Form: $y = mx + b$

  • Used when slope $m$ and y-intercept $b$ are known.

• Standard Form: $Ax + By = C$

• Algebraic Criteria:

  • Both $x$ and $y$ must be raised strictly to the first power ($x^1$, $y^1$).

  • Variables cannot appear in exponents, denominators, or radicals.

Horizontal and Vertical Lines

• Horizontal Lines:

  • Equation: $y = b$ (slope $m = 0$).

  • The x-axis is $y = 0$.

• Vertical Lines:

  • Equation: $x = c$ (slope is undefined/infinite).

  • Not functions (fail vertical line test).

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    Parallel and Perpendicular Lines

    • Interdependence: Horizontal and vertical lines are perpendicular ($90^\circ$).

    • Parallel Lines: Possess identical slopes ($m_1 = m_2$) and different y-intercepts ($b_1 \neq b_2$).

    • Perpendicular Lines: Intersect at $90^\circ$ with opposite reciprocal slopes ($m \rightarrow -\frac{1}{m}$).