2.3 Linear Functions, Slope & Rate of Change
Key Takeaways
- A linear function is characterized by a constant rate of change; its algebraic formula f(x) = mx + b generates a straight line where m represents the slope and b represents the y-intercept f(0).
- Slope measures the vertical change relative to horizontal change: m = (y2 - y1) / (x2 - x1) = Delta y / Delta x = Rise / Run.
- In real-world modeling contexts, slope represents the physical rate of change, expressed in units of 'output units per one input unit' (e.g., dollars per hour, miles per gallon).
- Parallel lines have identical slopes (m1 = m2) and distinct y-intercepts; perpendicular lines have negative reciprocal slopes (m1 * m2 = -1, or m2 = -1/m1).
- Horizontal lines have a slope of 0 and equations of the form y = c; vertical lines have an undefined slope and equations of the form x = c.
Linear Functions, Slope & Rate of Change
A linear function is a mathematical relationship between an independent variable $x$ (input) and a dependent variable $y = f(x)$ (output) characterized by a strictly constant rate of change. On the Digital SAT, linear functions are evaluated across algebraic equations, function notation, data tables, and coordinate graphs.
1. Linear Function Fundamentals & Function Notation
The standard algebraic rule for a linear function is written as:
where $m$ represents the slope (rate of change) and $b$ represents the $y$-intercept (the initial value $f(0)$).
Function Notation on the Coordinate Plane
Understanding function notation is critical for converting SAT problem statements into coordinate pairs:
- The statement $f(a) = b$ means that when the input $x = a$, the output $y = b$. This corresponds directly to the coordinate point $(a, b)$ on the graph of $y = f(x)$.
- If $f(2) = 7$ and $f(5) = 16$, the line passes through the points $(2, 7)$ and $(5, 16)$.
Recognizing Linearity from Tables of Values
A table of values represents a linear function if and only if the first differences in output divided by the differences in input remain constant:
| $x$ | $f(x)$ | $\Delta x$ | $\Delta f(x)$ | Rate of Change $\frac{\Delta f(x)}{\Delta x}$ |
|---|---|---|---|---|
| $1$ | $5$ | — | — | — |
| $3$ | $11$ | $+2$ | $+6$ | $\frac{6}{2} = 3$ |
| $7$ | $23$ | $+4$ | $+12$ | $\frac{12}{4} = 3$ |
| $10$ | $32$ | $+3$ | $+9$ | $\frac{9}{3} = 3$ |
Because the ratio $\frac{\Delta f(x)}{\Delta x} = 3$ is constant for all intervals, $f(x)$ is a linear function with slope $m = 3$.
2. The Slope Formula & The Four Orientations of Slope
Given any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a non-vertical line, the slope $m$ is defined as:
+-----------------------------------------------------------------------------+
| THE FOUR ORIENTATIONS OF SLOPE |
| |
| POSITIVE SLOPE (m > 0) NEGATIVE SLOPE (m < 0) ZERO SLOPE (m = 0) |
| y y y |
| | / | \ | |
| | / | \ | ----------- |
| | / | \ | |
| --------+------ x --------+------ x --------+------ x |
| Rises Left to Right Falls Left to Right Horizontal Line |
| |
| UNDEFINED SLOPE (m undefined) |
| y |
| | | |
| | | x = c |
| | | |
| --------+---+-- x |
| Vertical Line (Run = 0, division by zero) |
+-----------------------------------------------------------------------------+
Worked Example: Fraction Coordinate Slope Calculation
Problem: Find the slope of the line passing through $\left(-\frac{3}{4}, 5\right)$ and $\left(2, -\frac{1}{2}\right)$.
Divide the fractions by multiplying by the reciprocal:
3. Interpreting Slope in Real-World Contexts (Dimensional Analysis)
On the SAT, slope is rarely just an abstract number—it represents a physical rate of change. To interpret the meaning of slope accurately in any word problem, attach units of measurement to the numerator and denominator:
Examples of Contextual Slope Interpretations
| Function Model | Slope $m$ | Correct Contextual SAT Interpretation |
|---|---|---|
| $C(h) = 45h + 80$ ($C$ = cost in $, $h$ = hours) | $+45$ | "The cost increases by $45 for each additional hour of service." |
| $T(m) = -2.4m + 95$ ($T$ = temp in $^{\circ}\text{F}$, $m$ = minutes) | $-2.4$ | "The temperature decreases by $2.4^{\circ}\text{F}$ for each 1-minute increase in time." |
| $E(d) = 0.35d + 120$ ($E$ = elevation in m, $d$ = km walked) | $+0.35$ | "The hiker gains $0.35$ meters of elevation for every $1$ kilometer traveled." |
| $V(t) = -1500t + 24000$ ($V$ = car value in $, $t$ = years) | $-1500$ | "The vehicle depreciates by $1,500 per year." |
[!TIP] SAT Trigger Words for Slope: Look for words like per, each, every, rate of change, rate of increase, hourly charge, or unit cost. These words almost universally point directly to the slope $m$ in a linear model.
4. Parallel and Perpendicular Lines
The geometric relationship between two lines in the $xy$-plane is completely determined by their slopes:
+-----------------------------------------------------------------------------+
| PARALLEL VS. PERPENDICULAR LINES |
| |
| PARALLEL LINES PERPENDICULAR LINES |
| - Identical slopes: m1 = m2 - Negative reciprocal: |
| - Never intersect - m1 * m2 = -1 |
| - Same steepness & direction - m2 = -1 / m1 |
| - Intersect at 90° angle |
| y y |
| | / / | / |
| | / / | / \ |
| | / / | / \ |
| --------+/---/--- x --------+-------\- x |
| /| / | \ |
+-----------------------------------------------------------------------------+
Negative Reciprocal Rule for Perpendicular Slopes
Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular if and only if:
To find the negative reciprocal of a fraction:
- Flip the numerator and denominator (take reciprocal).
- Change the sign (positive becomes negative, negative becomes positive).
| Original Slope ($m_1$) | Perpendicular Slope ($m_2$) | Calculation Verification ($m_1 \cdot m_2$) |
|---|---|---|
| $\frac{3}{5}$ | $-\frac{5}{3}$ | $\left(\frac{3}{5}\right)\left(-\frac{5}{3}\right) = -1$ |
| $-4$ | $+\frac{1}{4}$ | $(-4)\left(\frac{1}{4}\right) = -1$ |
| $-\frac{7}{2}$ | $+\frac{2}{7}$ | $\left(-\frac{7}{2}\right)\left(\frac{2}{7}\right) = -1$ |
| $1$ | $-1$ | $(1)(-1) = -1$ |
| $0$ (horizontal line $y = k$) | Undefined (vertical line $x = c$) | Orthogonal axes intersect at $90^{\circ}$ |
Worked Example: Finding a Perpendicular Line Equation
Problem: Line $L_1$ passes through $(4, -1)$ and is perpendicular to the line with equation $6x + 2y = 15$. What is the equation of Line $L_1$ in slope-intercept form?
Step 1: Find the slope of the given line
Step 2: Determine the perpendicular slope
Step 3: Use point-slope form with point $(4, -1)$
Step 4: Convert to slope-intercept form Final Answer: $y = \frac{1}{3}x - \frac{7}{3}$.
5. Digital SAT Desmos Workflows for Linear Functions
Desmos makes evaluating, plotting, and regressing linear functions instantaneous:
- Defining Functions: Type
f(x) = -2x + 7. To evaluate $f(7)$, simply typef(7)on the next line to get-7immediately. - Table Creation & Rate of Change: Click the
+icon, select Table, and enter coordinate points. Desmos plots the points automatically. - Linear Regression for Slope: In a table with columns $x_1$ and $y_1$, type:
y_1 ~ m*x_1 + bDesmos will output the exact slope $m$ and $y$-intercept $b$ with $r^2 = 1$.
A linear function f satisfies f(-2) = 11 and f(4) = -1. What is the value of f(7)?
Line k in the xy-plane is perpendicular to the line with equation 3x + 5y = 20. If line k passes through the point (6, -1), which of the following is an equation of line k?
The height H(t), in feet, of a hot air balloon t minutes after beginning its descent is modeled by a linear function. At t = 4 minutes, the balloon is at a height of 1,620 feet, and at t = 9 minutes, the balloon is at a height of 1,270 feet. Which of the following statements is the best interpretation of the slope of the graph of H in the xy-plane?