Slope-Intercept Form
Read y = mx + b as a complete linear story: b gives the output when x starts at zero, and m tells how that output changes for each step in x.
How can two numbers describe an entire straight-line relationship?
Slope-intercept form separates a linear relationship into two jobs: a starting output and a constant rate of change from that start.
m is slope. b is the y-intercept, the value of y when x = 0.
Read y = 2x − 1 from the starting point outward.
Setting x to zero exposes the intercept. Then slope tells us what happens as x moves away from zero.
The line crosses the y-axis at (0, −1), so b = −1.
The slope m = 2 means every rightward step of 1 raises y by 2.
Repeating the same rate creates ordered pairs that all satisfy the equation.
The same equation coordinates the table, graph, and verbal rate. These are different views of one relationship.
The y-intercept is not merely where the line happens to cross. It is the output built into the rule when the input is zero.
Once the line is anchored, the slope determines the same change in y for every equal change in x.
Change one parameter at a time.
Isolating one parameter makes its job visible instead of turning the first interaction into a two-slider sandbox.
m stays at 2. Changing b slides the line to a new starting height without changing its steepness, so the resulting lines are parallel.
Three special readings are worth recognizing immediately.
The starting output is zero, so the line passes through the origin.
The output never changes as x changes, so the line is horizontal.
A vertical line has undefined slope, so it cannot be written as y = mx + b. A later lesson compares the forms that handle this cleanly.