Slope & Rate of Change
Slope compares how two quantities change together. Read it as a ratio, see it as a rise-and-run triangle, and interpret it in the units of the situation.
Slope is one ratio seen in several forms.
Change the rise, run, starting value, or point order. The graph, numerical ratio, verbal interpretation, and moving direction field remain synchronized.
As x increases, y increases. The line rises from left to right.
y = 2x + 1
Change the rate, slide the measurement triangle, or reverse the points. The graph, ratio, and background use the same state.
For every 1 unit x moves right, y increases by 2.
The table, graph, and context should tell the same change story.
For the relationship y = 1.5x + 1, every increase of 2 in x produces an increase of 3 in y. The slope is 3/2, or 1.5 output units per input unit.
Repeated differences
A repeated staircase
Any rise-and-run triangle drawn on the same line reduces to the same ratio.
Units complete the meaning
A tank gains 1.5 liters per minute. The ratio carries output units over input units.
Keep the subtraction order consistent in the numerator and denominator.
Reversing the points is legal. Mixing the orders is not: using y₂ − y₁ with x₁ − x₂ would change only one sign and produce the opposite slope.
The line rises left to right.
The line falls left to right.
y stays constant while x changes.
A vertical line would divide by zero.