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 $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$:

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

  • $b = y<em>1 - m x</em>1$ (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 $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$; solve for $b$ using $b = y<em>1 - m x</em>1$.

Quick Practice Steps

• Step 1: Identify two points on the line.

• Step 2: Compute $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

• Step 3: Compute $b = y<em>1 - m x</em>1$.

• 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 $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$?

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: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

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.