Section 2.lin

Linear Functions

Linear functions are guaranteed on the final — expect 2–3 questions on slopes, equations of lines, and parallel/perpendicular relationships. They're the simplest model for constant-rate-of-change situations and the baseline everything else gets compared to.

⚡ Quick Summary

A linear function is the simplest type of relationship: it grows (or shrinks) at a perfectly steady rate, like a car driving at a constant speed. On a graph, it's always a straight line — no curves, no surprises.

Slope

m = \frac{y_2 - y_1}{x_2 - x_1}

Rise over run — the rate of change between any two points

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The slope tells you how steep the line is — think of it as the steepness of a hill. A positive slope goes uphill (left to right), a negative slope goes downhill, and a zero slope is perfectly flat. You calculate it as "rise over run": how much the output changes divided by how much the input changes.

There are two handy formulas for writing the equation of a line. Slope-intercept form $y = mx + b$ is great when you know the slope and where the line crosses the $y$ -axis. Point-slope form is your go-to when you know the slope and any single point on the line. You can always convert between them.

Two lines are parallel if they have the exact same steepness (same slope) — they run side by side and never touch. Two lines are perpendicular if they meet at a right angle, which happens when their slopes are negative reciprocals — flip the fraction and change the sign.

Key Formulas

Slope

m = \frac{y_2 - y_1}{x_2 - x_1}

Rise over run — the rate of change between any two points

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept

Point-Slope Form

y - y_1 = m(x - x_1)

Use when you know the slope and one point $(x_1, y_1)$

Parallel & Perpendicular Slopes

\text{Parallel: } m_1 = m_2 \qquad \text{Perpendicular: } m_1 \cdot m_2 = -1

Perpendicular slopes are negative reciprocals (e.g. $\frac{2}{3}$ and $-\frac{3}{2}$ )

Key Takeaways

✓Slope $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ — positive means uphill, negative means downhill, zero means horizontal
✓Point-slope form $y - y_1 = m(x - x_1)$ is the fastest way to write a line equation when you have a point and slope
✓Parallel lines have equal slopes ( $m_1 = m_2$ ); perpendicular lines have negative reciprocal slopes ( $m_1 \cdot m_2 = -1$ )

⚠️ Common Mistakes

✗Finding the perpendicular slope by flipping the fraction but forgetting to change the sign — perpendicular to $\frac{2}{3}$ is $-\frac{3}{2}$ , not $\frac{3}{2}$
✗Putting the slope formula upside down — it's $\frac{y_2 - y_1}{x_2 - x_1}$ (rise over run), not $\frac{x_2 - x_1}{y_2 - y_1}$
✗Distributing incorrectly in point-slope form — in $y - y_1 = m(x - x_1)$ , multiply $m$ by BOTH the $x$ and the $-x_1$ inside the parentheses
✗Confusing horizontal and vertical lines — horizontal lines have slope $0$ (equation $y = c$ ), vertical lines have undefined slope (equation $x = c$ )

Next: Section 3.cq Builds on: 1.fn

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1.inv Inverse Functions

3.cq Complex Numbers & Quadratics