Slope–Intercept Form Comprehensive Notes

Review of the Slope–Intercept Form

The lesson centers on the fundamental linear‐equation format $y = mx + b$, where each symbol carries a specific geometric meaning on the Cartesian plane.

• $x, y$ – independent and dependent variables (ordered as $(x, y)$) that locate points.
• $m$ – the slope: a constant rate of change, visualised as line “steepness” or “tilt.”
  • Computed by either verbal “rise over run” or the formal two-point formula $m = \dfrac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.
  • Positive $m$ rises left-to-right; negative $m$ falls.
• $b$ – the $y$-intercept: the point where the line crosses the vertical axis.
  • The associated ordered pair is $(0, b)$ because $x = 0$ on the $y$-axis.

Core Vocabulary & Concepts

Axes & Orientation

• $x$-axis – horizontal (left/right).
• $y$-axis – vertical (up/down).
• Label axes before extracting intercepts; mis-labelling swaps coordinates.

Finding the $y$-Intercept on a Graph

1. Locate the vertical $y$-axis.
2. Identify where the plotted line intersects it.
3. Read the $y$ coordinate; $x$ will be $0$.
   • Example: A line crossing at $6$ gives $b = 6$ and intercept $(0, 6)$.

Illustrative Q&A: “Which ordered pair describes the intercept at $y = 3$?”
Correct choice: $(0, 3)$.

Writing Equations Given $m$ and $b$

Example parameters: $m = -3$, $b = -9$.
Acceptable equivalent forms:

• $y = -3x + (-9)$
• $y = -3x - 9$
• $y = -9 + (-3x)$
  All satisfy the template $y = mx + b$ despite term order.

Additional example: $m = 7$, passes through $(0, 4)$ → $b = 4$.
Equation: $y = 7x + 4$.

Deriving $y = mx + b$ from a Graph

Process demonstrated twice:

1. Compute slope using two lattice steps (each square = 1 unit).
   • Rise $= 2$, Run $= 1$ → $m = \frac{2}{1} = 2$.
2. Read $y$-intercept directly; in example, $b = 1$ ($(0,1)$).
3. Substitute: $y = 2x + 1$.

Second graph:

• Rise $= 4$, Run $= 2$ → $m = \frac{4}{2} = 2$.
• Intercept $(0,3)$ → $b = 3$.
• Equation: $y = 2x + 3$.

Constructing Equations from Tables

Case 1: Table already lists $x = 0$

Example table

• Differences: $\Delta y = 4$, $\Delta x = 2$ → $m = \frac{4}{2} = 2$.
• $y$-Intercept appears explicitly: $(0,4)$ → $b = 4$.
  Equation: $y = 2x + 4$.

Case 2: Table missing $x = 0$

Example table

1. Still compute slope via consecutive rows.
   • $\Delta y = 4$, $\Delta x = 2$ → $m = 2$.
2. Plug one data point $(2,6)$ into $y = mx + b$ to solve for $b$:
   $6 = 2(2) + b \Rightarrow 6 = 4 + b \Rightarrow b = 2.$
3. Final equation: $y = 2x + 2$.

Second practice problem: smallest point $(4,6)$, slope previously found $m = 3$.
$6 = 3(4) + b \Rightarrow 6 = 12 + b \Rightarrow b = -6$ ⇒ $y = 3x - 6$.

Word-Problem Application

Scenario: “John makes $\$40$ daily plus $\$10$ per bag packed.”

• Interpretation:
  • Fixed daily pay → $y$-intercept $b = 40$.
  • Variable pay per bag → slope $m = 10$ (dollars/bag).
• Independent variable $x$ = number of bags; dependent $y$ = total daily earnings.
• Model: $y = 10x + 40$.
  • When $x = 0$ bags, $y = 40$—matches base salary.
  • Each additional bag increments earnings by $\$10$, reflecting the constant rate.

Alternative Valid Algebraic Forms

Although $y = mx + b$ is canonical, rearranging terms or using additive inverses preserves equality:

• $y - mx = b$
• $mx - y = -b$
• $b = y - mx$

All describe the identical line provided coefficients are unchanged.

Key Formulas & Numerical References

1. Slope (two-point): $m = \dfrac{y<em>2-y</em>1}{x<em>2 - x</em>1}$
2. Slope-Intercept Structure: $y = mx + b$
3. Ordered-pair format: $(x, y)$ with $x$ first, $y$ second.

Step-by-Step Procedures

A. From a Graph

1. Count a consistent rise/run between any two lattice points.
2. Reduce the fraction if possible to simplest $m$.
3. Identify intercept $(0,b)$.
4. Substitute into $y = mx + b$.

B. From a Table

1. Compute $\Delta y$ and $\Delta x$ for consecutive rows.
2. Calculate $m = \Delta y / \Delta x$.
3. If $(0,b)$ appears, read $b$; otherwise solve $b$ using any point.

C. From a Verbal Description

1. Fixed starting amount → $b$.
2. “Per, each, every” rate → $m$.
3. Let $x$ represent the counted item, write $y = mx + b$.

Ethical/Practical Implications

• Linear models simplify complex realities; ensure constant-rate assumptions truly hold (e.g.
  John’s pay might cap at overtime, violating linearity).
• Graphical clarity: Always label axes to avoid communicative errors in multicultural or interdisciplinary settings.

Connections to Prior & Future Lessons

• Builds directly upon previous slope-only lesson (rise/run triangle, lattice counting).
• Prepares ground for next topics:
  • Graphing full lines using intercept/slope.
  • Converting between slope-intercept, point-slope, and standard forms.
  • Solving linear systems and interpreting real-world linear relationships.