Linear relationships
One quantity changes at a steady rate as another grows. Why is the graph a straight line, and how many ways can you write its equation?
Where you are 11 inside
What this area is about
A linear relationship is one where the output changes by the same amount for every unit step in the input. Its graph is a straight line, and its rate of change is the line’s slope. This area is about reading that line and its equation in each of their usual forms, and moving between them.
How these fit together
The rate. Slope is rise over run, the same between any two points of a non-vertical line. Direct variation is the case y = kx with k \ne 0: a line through the origin.
Forms of the equation. Slope-intercept form, y = mx + b, shows the slope and the y-intercept. Point-slope form starts from any point, or from two points. Standard form, Ax + By = C, fits a fixed total shared by two quantities.
Graphing. From a table of values, point by point, and from the equation, intercept then slope.
Two lines. Parallel lines that are not vertical share a slope; perpendicular lines have slopes whose product is -1, when neither line is vertical. Writing a parallel or perpendicular line puts both to use.
Using it. Modeling with linear relationships.
A path through it
Start with slope and with graphing from a table. After slope: slope-intercept form and direct variation. Then graphing from the equation, and point-slope form. After point-slope form: standard form (which also needs the table) and parallel lines; then perpendicular lines, then writing either kind of line. Modeling comes after direct variation.
After this: systems of equations, the line of best fit and sequences.