2.3 Linear Functions, Slope & Rate of Change

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.

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1. Linear Function Fundamentals & Function Notation

The standard algebraic rule for a linear function is written as:

$$f(x) = mx + b$$

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:

$$\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \text{constant } m$$

Because the ratio $\frac{\Delta f(x)}{\Delta x} = 3$ is constant for all intervals, $f(x)$ is a linear function with slope $m = 3$.

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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:

$$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{Change in Vertical Position (Rise)}}{\text{Change in Horizontal Position (Run)}}$$

+-----------------------------------------------------------------------------+
|                        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 $\left(2, -\frac{1}{2}\right)$.

$$m = \frac{-\frac{1}{2} - 5}{2 - \left(-\frac{3}{4}\right)} = \frac{-\frac{1}{2} - \frac{10}{2}}{\frac{8}{4} + \frac{3}{4}} = \frac{-\frac{11}{2}}{\frac{11}{4}}$$

Divide the fractions by multiplying by the reciprocal:

$$m = -\frac{11}{2} \times \frac{4}{11} = -\frac{44}{22} = -2$$

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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:

$$\text{Unit of Slope } m = \frac{\text{Units of Dependent Variable } (y)}{\text{Units of Independent Variable } (x)}$$

Examples of Contextual Slope Interpretations

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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:

$$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$$

To find the negative reciprocal of a fraction:

1. Flip the numerator and denominator (take reciprocal).
2. Change the sign (positive becomes negative, negative becomes positive).

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

$$2y = -6x + 15 \implies y = -3x + \frac{15}{2} \implies m_1 = -3$$

Step 2: Determine the perpendicular slope

$$m_2 = -\frac{1}{-3} = \frac{1}{3}$$

Step 3: Use point-slope form with point $(4, -1)$

$$y - (-1) = \frac{1}{3}(x - 4) \implies y + 1 = \frac{1}{3}x - \frac{4}{3}$$

Step 4: Convert to slope-intercept form

$$y = \frac{1}{3}x - \frac{4}{3} - 1 = \frac{1}{3}x - \frac{7}{3}$$

Final Answer: $y = \frac{1}{3}x - \frac{7}{3}$.

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5. Digital SAT Desmos Workflows for Linear Functions

Desmos makes evaluating, plotting, and regressing linear functions instantaneous:

1. Defining Functions: Type f(x) = -2x + 7. To evaluate $f(7)$, simply type f(7) on the next line to get -7 immediately.
2. Table Creation & Rate of Change: Click the + icon, select Table, and enter coordinate points. Desmos plots the points automatically.
3. Linear Regression for Slope: In a table with columns $x_1$ and $y_1$, type: y_1 ~ m*x_1 + b Desmos will output the exact slope $m$ and $y$-intercept $b$ with $r^2 = 1$.