Slope-Intercept Form: Quick Reference

Slope-Intercept Form

  • Definition: y = mx + b where m is the slope and b is the y-intercept.

  • Slope-intercept form used to describe a line concisely.

Reading from a Graph

  • Y-intercept: the value of y where the line crosses the y-axis (set x = 0), equals b.

  • Slope: rise over run; m = \frac{\Delta y}{\Delta x}.

  • Positive slope: line rises to the right; Negative slope: line falls to the right.

Finding m and b from Points

  • Given two points (x1, y1) and (x2, y2) :

    • m = \frac{y2 - y1}{x2 - x1}

    • b = y1 - m x1 (or use the second point)

  • Then write the equation: y = mx + b.

Worked Examples from the Transcript

  • Slope 1, y-intercept 3: y = x + 3

  • Slope 2, y-intercept 2: y = 2x + 2

  • Slope -1, y-intercept 6: y = -x + 6

  • Slope -3, y-intercept 7: y = -3x + 7

Quick Graph Verification

  • Use graphing tool (Desmos) to graph the equation and compare with the paper diagram.

  • If you pick two points on the line, the calculated m should match the visual slope; the intercept should match the crossing on the y-axis.

Quick Rules for Quick Recall

  • If the line passes through the point at x = 0 with y = b, that is the y-intercept in the equation y = mx + b.

  • When slope magnitude > 1 (e.g., m = 2), for each 1 unit of run, rise is 2 units; you can also think as rise 2 over run 1.

  • To verify a line: pick two points on the line and compute m = \frac{y2 - y1}{x2 - x1}; solve for b using b = y1 - m x1.

Quick Practice Steps

  • Step 1: Identify two points on the line.

  • Step 2: Compute m = \frac{y2 - y1}{x2 - x1}.

  • Step 3: Compute b = y1 - m x1.

  • Step 4: Write final equation y = mx + b.

  • Step 5: If using Desmos, type y = mx + b and adjust to match the graph.

Knowledge Test

  1. What is the definition of slope-intercept form?

  2. How do you calculate the slope (m) of a line given two points (x1, y1) and (x2, y2) ?

  3. If a line has a positive slope, which direction does it rise on a graph?

  4. Find the equation of a line (y = mx + b) that passes through the points (1, 5) and (3, 9) . Show your work.

  5. What value in the slope-intercept form represents the y-intercept?

Answer Key

  1. Slope-intercept form is defined as y = mx + b, where m is the slope and b is the y-intercept.

  2. The slope (m) is calculated using the formula: m = \frac{y2 - y1}{x2 - x1}.

  3. If a line has a positive slope, it rises to the right.

  4. Given points (1, 5) and (3, 9) :

    • Compute slope (m): m = \frac{9 - 5}{3 - 1} = \frac{4}{2} = 2

    • Compute y-intercept (b): Using point (1, 5) , b = 5 - (2)(1) = 5 - 2 = 3

    • The equation is: y = 2x + 3

  5. The value b in the equation y = mx + b represents the y-intercept.