Slope Notes

Definition & Core Idea of Slope

Slope measures the rate of change between two variables in a linear relationship.

• Symbol: $m$ (commonly used in algebra).

• Verbal mnemonic: "rise over run" – how much the graph rises (vertical change) for every unit it runs (horizontal change).

• Formal ratio: $m = \frac{\text{change in }y}{\text{change in }x} = \frac{\Delta y}{\Delta x}$, where $\Delta$ (Greek capital delta) means “change in.”

• Geometric meaning: the steepness or inclination of a straight line; it stays constant for any two points on that line.

Visual Interpretation on Coordinate Graphs

When traced on a grid, the points often look like stair-steps you can follow from one lattice point to the next.

• "Rise" = how many units you move upward (positive) or downward (negative) along the $y$-axis.

• "Run" = how many units you move rightward (positive) or leftward (negative) along the $x$-axis.

Categories of Slopes and Their Graph Shapes

Mnemonic image: “P N Z U” – Positive (up-right), Negative (down-right), Zero (horizontal), Undefined (vertical).

Procedure: Finding Slope Directly From a Graph

1. Pick any two lattice points on the line (points whose coordinates are integers).

2. Starting at the first point:
   • Count the vertical change (rise).
   • Count the horizontal change (run) required to land on the second point.

3. Apply $\displaystyle m = \frac{\text{rise}}{\text{run}}$.

4. Simplify the fraction (reduce common factors, move the negative sign to the numerator if you prefer $m= -\tfrac{a}{b}$).

Because a line’s slope is constant, any pair of distinct points will yield the same $m$.

Worked Graph Examples

Example 1 (Positive)
Graph passes (1,1) then goes up 3, right 6: $m = \tfrac{3}{6}=\tfrac12$. Using a smaller stair-step between nearer lattice points (up 1, right 2) gives the same $\tfrac12$.

Example 2 (Negative)
From a point you can read down 1, right 2 → $m = \tfrac{-1}{2}$. Equivalently up 1, left 2 → $m = \tfrac{1}{-2}$ – the same value.

Example 3 (Zero slope)
Horizontal line: rise $=0$, run $=3$ ⇒ $m = \tfrac{0}{3}=0$.

Example 4 (Undefined)
Vertical line: rise $=3$, run $=0$ ⇒ $m = \tfrac{3}{0}$ (division by zero), so undefined.

Algebraic Method: Slope Formula for Two Points

Given ordered pairs $P<em>1(x</em>1, y<em>1)$ and $P</em>2(x<em>2, y</em>2)$:
$\boxed{\displaystyle m = \frac{y<em>{2}-y</em>{1}}{x<em>{2}-x</em>{1}}}$

Steps

1. Label each coordinate (first point $x<em>1,y</em>1$ ; second $x<em>2,y</em>2$).

2. Subtract $y$’s (top) and $x$’s (bottom).

3. Simplify, watch signs.

Additive-Inverse Shortcut: Turning “subtract a negative” into “add a positive.”

Detailed Worked Examples

1. Points $(-6,2)$ and $(9,-3)$
   $\begin{aligned}<br>m &amp;= \frac{-3-2}{9-(-6)} \ &amp;= \frac{-5}{15}=\frac{-1}{3}<br>\end{aligned}$

2. Points $(5,-5)$ and $(7,3)$
   $m = \frac{3-(-5)}{7-5}=\frac{8}{2}=4$ (or $\tfrac41$).

3. Points $(2,6)$ and $(-4,2)$
   $m = \frac{6-2}{2-(-4)}=\frac{4}{6}=\tfrac23$.

4. Points $(3,-10)$ and $(2,1)$
   $m = \frac{1-(-10)}{2-3}=\frac{11}{-1}=-11$.

5. Points $(1,9)$ and $(3,8)$
   $m = \frac{8-9}{3-1}=\frac{-1}{2}$ → negatives cancel when both numerator and denominator negative.

Slope from Tables (Tabular Data)

A table implicitly lists points $(x,y)$. Compute successive changes (differences):

1. Find $\Delta y$ (how much y increases or decreases each step).

2. Find $\Delta x$ (how much x increases or decreases simultaneously).

3. $m = \tfrac{\Delta y}{\Delta x}$ should be constant for every row-to-row jump if the relationship is linear.

4. Optionally verify with the two-point formula using any pair from the table.

Table Example A

Differences: $\Delta y = +1$, $\Delta x = +2$ ⇒ $m = \tfrac12$.
Check with endpoints: $m = \tfrac{5-3}{2-(-2)}=\tfrac{2}{4}=\tfrac12$.

Table Example B

Differences: $\Delta y = +7$, $\Delta x = +1,$ so $m=7$. Using bottom two points: $m=\tfrac{70-49}{10-7}=\tfrac{21}{3}=7$ – consistent.

Real-World Interpretation & Word Problems

Slope often answers “How much per unit?”

Key linguistic cues: each, every, per.

• If both variables move the same direction (both increase or both decrease), m>0.
• If one increases while the other decreases, m<0.

Example Scenario

“You’re charged $\$63$ for every 9 games, plus an additional $\$8$ fee.”

• Variable cost (slope): $m = \tfrac{63}{9}=7$ dollars per game.
• Fixed fee = $8$ dollars (y-intercept, to be used in later lessons on linear equations).

Practical use: with the cost model $\text{Total} = 7x + 8$ you could predict price for 5, 10, or 30 games once you’ve mastered full linear‐function form.

Ethical, Philosophical & Pedagogical Notes

• Recognizing slope as “change over change” is foundational for calculus (derivative) and socioeconomic rate studies (inflation‐per‐year, population‐growth-per-decade).
• Undefined slopes (vertical lines) remind us that not every relationship is a function (fails vertical‐line test).
• Graphical intuition (“stairs” metaphor) bridges algebraic formulas with visual learning styles.

Summary

• Formula: m=x2−x1y2−y1​=ΔxΔy​.
• Positive ↗, Negative ↘, Zero (horizontal), Undefined (vertical).
• On graphs: COUNT boxes; in tables: FIND differences; with coordinates: SUBSTITUTE into formula.
• Always reduce the fraction; keep the negative sign in the numerator or in front.
• In contexts: slope = “$\text{output change}\big/\text{input change}$.”