8.2 Linear Functions: Slope, Rate of Change, & Linear Equations
Key Takeaways
- The slope ($m$) of a linear function measures its constant rate of change, calculated algebraically as $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ ('rise over run'), representing the change in output per unit increase in input.
- The four geometric categories of slope are positive (rising from left to right), negative (falling from left to right), zero (horizontal line $y = c$), and undefined (vertical line $x = c$, which is not a function).
- Linear equations are represented in three standard forms: Slope-Intercept Form ($y = mx + b$), Point-Slope Form ($y - y_1 = m(x - x_1)$), and Standard Form ($Ax + By = C$, where intercepts are found by setting $y = 0$ and $x = 0$).
- Parallel lines have identical slopes ($m_1 = m_2$) with different intercepts, whereas perpendicular lines have negative reciprocal slopes ($m_1 \cdot m_2 = -1$ or $m_2 = -\frac{1}{m_1}$) intersecting at right angles.
8.2 Linear Functions: Slope, Rate of Change, & Linear Equations
Linear functions represent the most widely used mathematical models on the TABE 13&14 Mathematics assessment and across technical trades. A linear relationship describes any scenario where a dependent variable changes at a constant, steady rate relative to an independent variable—such as steady vehicle fuel consumption, constant hourly wage accumulation, or linear thermal expansion in piping.
1. Slope as a Constant Rate of Change
The defining characteristic of any linear function is that its rate of change is constant. On a coordinate plane, this constant rate of change is called the slope ($m$) of the line. Geometrically, slope quantifies the steepness and direction of a line, defined as the ratio of vertical change ("rise") to horizontal change ("run"):
Step-by-Step Algebraic Slope Calculation
To calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$:
- Assign coordinates carefully: let $(x_1, y_1) = (-4, 7)$ and $(x_2, y_2) = (6, -8)$.
- Substitute into the slope formula:
- Workplace Interpretation: For every increase of $2$ units in $x$, the quantity $y$ decreases by $3$ units.
2. The Four Classifications of Slope
Every straight line in coordinate geometry falls into one of four distinct slope categories:
Four Geometric Slope Classifications:
1. POSITIVE (m > 0) 2. NEGATIVE (m < 0) 3. ZERO (m = 0) 4. UNDEFINED (m = undefined)
/ \ |
/ \ ─────────────────── |
/ \ |
Rises L -> R Falls L -> R Horizontal (y = c) Vertical (x = c)
| Slope Category | Value Condition | Geometric Behavior | Equation Form | Function Status |
|---|---|---|---|---|
| Positive Slope | $m > 0$ | Rises from left to right (increasing function) | $y = mx + b$ ($m > 0$) | Valid Function |
| Negative Slope | $m < 0$ | Falls from left to right (decreasing function) | $y = mx + b$ ($m < 0$) | Valid Function |
| Zero Slope | $m = 0$ | Perfectly horizontal; $\Delta y = 0, \Delta x \neq 0$ | $y = c$ (constant value) | Valid Function |
| Undefined Slope | $\frac{\Delta y}{0}$ | Perfectly vertical; division by zero ($\Delta x = 0$) | $x = c$ (fixed input) | NOT a Function (Fails VLT) |
[!NOTE] Horizontal vs. Vertical Lines:
- A horizontal line has an equation like $y = 4$. Its slope is $0$. It passes the Vertical Line Test and is a constant function.
- A vertical line has an equation like $x = -3$. Its slope is undefined (division by zero). It fails the Vertical Line Test because one input has infinitely many outputs.
3. The Three Forms of Linear Equations
Depending on the given information (slope, points, intercepts), linear equations can be written in three algebraic forms:
A. Slope-Intercept Form: $y = mx + b$
- Components: $m = \text{slope}$, $b = y\text{-intercept}$ (the point $(0, b)$ where the graph crosses the vertical $y$-axis).
- Best used when: The rate of change and starting value are known, or when graphing directly.
B. Point-Slope Form: $y - y_1 = m(x - x_1)$
- Components: $m = \text{slope}$, $(x_1, y_1) = \text{any specific point on the line}$.
- Best used when: Given a slope and a single point, or when calculating a line through two given points.
C. Standard Form: $Ax + By = C$
- Components: $A, B, C$ are integers (with $A \ge 0$, and $A, B$ not both zero).
- Best used when: Finding coordinate intercepts quickly ($x$-intercept and $y$-intercept) or modeling combination constraints (e.g., $5x + 3y = 60$).
Quick Method: Intercepts from Standard Form
To graph $Ax + By = C$ using intercepts:
- $x$-intercept (set $y = 0$): $Ax + B(0) = C \implies x = \frac{C}{A} \implies \left(\frac{C}{A}, 0\right)$.
- $y$-intercept (set $x = 0$): $A(0) + By = C \implies y = \frac{C}{B} \implies \left(0, \frac{C}{B}\right)$.
Example: Find the intercepts of $4x - 5y = 40$.
- Set $y = 0$: $4x = 40 \implies x = 10 \implies (10, 0)$.
- Set $x = 0$: $-5y = 40 \implies y = -8 \implies (0, -8)$.
4. Converting Among Linear Forms (Step-by-Step)
Mastering algebraic transitions between forms is a core TABE skill:
Worked Demonstration: Find the equation of the line passing through $(2, -3)$ and $(6, 5)$ in slope-intercept form and standard form.
- Calculate Slope ($m$):
- Apply Point-Slope Form using $(2, -3)$:
- Convert to Slope-Intercept Form ($y = mx + b$):
- Convert to Standard Form ($Ax + By = C$):
5. Parallel vs. Perpendicular Lines
The geometric relationship between two straight lines is governed entirely by their slopes:
Line Relationships:
├── Parallel Lines ─────────> Identical Slopes: m1 = m2 (Different intercepts b1 ≠ b2)
└── Perpendicular Lines ────> Negative Reciprocal Slopes: m1 · m2 = -1 (m2 = -1 / m1)
| Line Relationship | Slope Condition | Intercept Condition | Geometric Behavior |
|---|---|---|---|
| Parallel Lines | $m_1 = m_2$ | $b_1 \neq b_2$ | Lines run in the same direction and never intersect |
| Perpendicular Lines | $m_2 = -\frac{1}{m_1}$ or $m_1 \cdot m_2 = -1$ | Any intercepts | Lines intersect at a precise $90^\circ$ right angle |
| Coincident Lines | $m_1 = m_2$ | $b_1 = b_2$ | Same identical line; infinitely many intersections |
Worked Example: Finding Perpendicular Equations
Problem: Find the slope-intercept equation of the line that passes through $(4, -1)$ and is perpendicular to the line $2x - 3y = 9$.
- Find slope of the given line:
- Determine perpendicular slope ($m_2$):
- Use Point-Slope Form with $(4, -1)$: y - (-1) &= -\frac{3}{2}(x - 4) \\ y + 1 &= -\frac{3}{2}x + \left(-\frac{3}{2}\right)(-4) \\ y + 1 &= -\frac{3}{2}x + 6 \\ y &= -\frac{3}{2}x + 5 \end{aligned}$$
6. Applied Linear Modeling: Workplace Word Problems & Tables
In practical applications, linear functions model scenarios with a fixed initial amount (the $y$-intercept $b$) plus a variable rate per unit (the slope $m$):
Worked Workplace Problem: Industrial Tank Drainage
Problem: An industrial water tank is drained at a constant rate. A telemetry sensor records the remaining volume: at $t = 3\text{ hours}$, volume is $4,100\text{ gallons}$; at $t = 8\text{ hours}$, volume is $2,750\text{ gallons}$.
- Find Rate of Drainage (Slope):
- Find Initial Tank Capacity ($b$):
- Complete Linear Model: $V(t) = -270t + 4910$.
- Time to Empty Tank ($V(t) = 0$):
What is the equation in slope-intercept form of the line that passes through the point (4, -1) and is perpendicular to the line 2x - 3y = 9?
An industrial water tank is drained at a constant rate. A sensor records the remaining water volume: at t = 3 hours, the volume is 4,100 gallons; at t = 8 hours, the volume is 2,750 gallons. Which linear function V(t) models the remaining volume after t hours, and what was the initial volume of water in the tank?
What are the coordinate intercepts of the linear equation 4x - 5y = 40?