Building from a Point
What if you don't know the y-intercept, but you know the slope and one random point \((x_1, y_1)\)? We use Point-Slope Form.
\[y - y_1 = m(x - x_1)\]
Worked Examples
Example 1: Building an Equation
Find the equation of a line with slope 3 that passes through \((4, 2)\).
- \(m=3, x_1=4, y_1=2\).
- Equation: \(y - 2 = 3(x - 4)\).
- Simplify: \(y - 2 = 3x - 12 \to y = 3x - 10\).
- Result: \(y = 3x - 10\)
Example 2: Two-Point Problem
Find the equation through \((1, 5)\) and \((3, 9)\).
- Step 1: Find slope. \(m = (9-5)/(3-1) = 4/2 = 2\).
- Step 2: Use Point-Slope with either point. \(y - 5 = 2(x - 1)\).
- Simplify: \(y - 5 = 2x - 2 \to y = 2x + 3\).
- Result: \(y = 2x + 3\)
The Bridge to Quantum Mechanics
In experimental physics, we never have the "perfect" equation starting at zero. We have raw data points. Point-Slope form is the tool used for Linear Regression—finding the best-fit line through experimental data. When we measure the behavior of a quantum semiconductor, we take two points of data and use this algebra to build the model of the material's energy gap. This is how you turn a laboratory measurement into a physical theory.