Why Your Slope Answers Are Always Wrong
Still chanting rac{y_2 - y_1}{x_2 - x_1} like a magic spell?
That's like trying to learn English by memorizing a dictionary. Most students mess up slope problems because they stare at a bunch of numbers but forget what slope actually feels like: Steepness.
Stop memorizing. Imagine a Ski Mountain instead.
If you view the coordinate plane as a Ski Resort, everything becomes instantly obvious.
The Core Rule: Ski Resort Guide
On this mountain, we ALWAYS move from Left to Right (just like reading a sentence). If you remember this, you will never mix up positive and negative signs again.
The Chairlift
Scenario: You are sitting on the lift, going UP to the peak.
Physics: You are gaining altitude against gravity.
Real World: Profit, growth, rocket trajectory.
Mnemonic: Positive = Pulling Up.
The Ski Run
Scenario: You are shredding down a perfect powder run.
Physics: Gravity is doing the work. Altitude drops.
Real World: Car depreciation, water flowing downhill.
Mnemonic: Negative = Nice ride down.
Cross-Country
Scenario: You are stuck on the flat lodge floor. No gravity, no lift.
Math Truth: .
Mnemonic: Zero Fun.
The Cliff
Scenario: This isn't a ski run; it's a vertical drop. If you try to ski this... game over.
Math Truth: You cannot divide by zero. It is impossible.
Mnemonic: Survival chance = Undefined.
When Will I Ever Use This?
Your Roof Might Collapse (Roof Pitch)
Architects don't usually call it "slope"—they call it Pitch. If you build a house in snowy Canada, you need a steep pitch (like 6/12, rising 6 inches for every 12 inches of run) so the snow slides off. If you build a flat roof (Zero Slope) in a snowstorm? Tons of snow will pile up and crush your living room.
The Treadmill "Calorie Burner"
When you set your treadmill to "Incline 5", you are setting a 5% slope. This means for every 100 meters you walk, you climb 5 vertical meters. Even changing from 0% to 1% drastically changes your calorie burn. That is calculation in action.
The Ultimate Formula
Now that you understand the "Why", the formula should look much friendlier:
Going Up (+) or Down (-)?
Always moving Right (+)