Slope & Linear Graphs: Rise Over Run, Three Equation Forms
Slope is rate of change — and rate of change is the entire point of Algebra 1. Master the slope formula, the three line equations (slope-intercept, point-slope, standard), and when to use each.
If a single concept dominates Algebra 1, it's slope. Every “rate” word problem, every linear function, every parallel-or-perpendicular question hinges on it. Slope is just one number that captures how fast y changes when x changes.
m = (y2 − y1) / (x2 − x1) = rise / runPick any two points on the line. Subtract y's, subtract x's, divide.
The four flavors of slope
Going up → positive. Going down → negative. Flat → zero. Vertical → undefined (you'd be dividing by zero).
Slope of a horizontal line
A horizontal line has slope:
📌 Horizontal line: y = constant. No rise. Slope = 0.
Slope of a vertical line
Slope of x=5?
📌 x=5 is vertical line. Slope is undefined.
The three equation forms
Three forms, same line. Use whichever matches what you're given.
Worked example: find the y-intercept
A line passes through (0, −2) and (4, 6). Find the y-intercept.
Read the points
The y-intercept is where the line crosses the y-axis — meaning x = 0. Look at your points: (0, −2). The y-intercept is the y-value when x = 0. Answer: −2. No formula needed.
Find the y-intercept
A line passes through (0, -2) and (4, 6). What is the y-intercept?
The y-intercept is the y-value when x = 0. The point (0, -2) is on the line, so y-intercept = -2.
Building an equation from scratch
Slope = −1 through (0, 3). Want the equation.
y = mx + by = (−1)x + 3y = −x + 3When the point is the y-intercept, slope-intercept form is fastest.
Write the equation
Find equation: slope=−1, through (0,3).
📌 y=−x+3 (y-intercept is 3).
Parallel & perpendicular lines
Two slope rules
Parallel lines → same slope (m1 = m2). Perpendicular lines → slopes are negative reciprocals (m1 · m2 = −1).
3-second recap
Slope = rise / run = (Δy) / (Δx)
Horizontal → m = 0; vertical → m undefined
y = mx + b: m is slope, b is y-intercept
y − y₁ = m(x − x₁): use when you have a point + slope
Parallel → same slope; perpendicular → negative reciprocal slopes
Check yourself
Quick check #1
Find the slope of the line through points (2, 5) and (6, 13).