Calculate and interpret the slope of a linear function as a rate of change from a graph, table, equation, or two points, and identify the meaning of slope and intercepts in context (A.F.3).

What this topic is asking

A.F.3 asks you to calculate and interpret slope as a rate of change and to read slope and intercepts from a graph, table, equation, or two points. On the Virginia Algebra I SOL these are Functions items: find a slope, interpret it in context, or identify intercepts. They appear as fill-in-the-blank, multiple choice, and coordinate-plane items.

Slope as rate of change

For a linear function, the slope is the constant rate of change: the amount $y$ changes for each $1$-unit increase in $x$. The formula is

Subtract the $y$-coordinates and the $x$-coordinates in the same order. For $(1, 4)$ and $(5, 16)$: $m = \frac{16 - 4}{5 - 1} = \frac{12}{4} = 3$.

Reading slope from each representation

• From a graph. Pick two points where the line crosses grid intersections and count rise over run (vertical change over horizontal change). Up-to-the-right is positive; down-to-the-right is negative.
• From a table. For a linear function the rate is constant, so divide any change in $y$ by the matching change in $x$. If $x$ increases by $2$ and $y$ increases by $6$, the slope is $3$.
• From an equation. In slope-intercept form $y = mx + b$, the coefficient $m$ is the slope and $b$ is the $y$-intercept.

Interpreting slope and intercepts in context

The biggest Functions skill here is translating slope and intercept into the situation:

• The slope is a rate: cost per item, miles per hour, dollars per week. Its units are output-units per input-unit.
• The $y$-intercept $b$ is the starting value: the output when the input is $0$ (a flat fee, an initial amount, a starting height).

For $C = 0.10t + 25$ (cost vs minutes), the slope $0.10$ is \$0.10 per minute** and the intercept$25 is the **\25 flat fee.

Why slope is constant only for lines

The defining feature of a linear function is that its rate of change is the same everywhere, and this is exactly what makes the slope a single number. Between any two points on a line, rise over run gives the identical value, which is why you can pick any two points and get the same slope. This constant rate is what a straight line looks like: equal steps up (or down) for equal steps across. Other function families do not share this: a quadratic speeds up its rate of change as you move along it, and an exponential multiplies rather than adds, so neither has one slope. That is why "rate of change" for a curve has to be measured over an interval (the average rate of change) rather than read off as a single slope. Recognizing that a constant rate of change is the signature of linearity helps you tell a linear table (constant differences in $y$ for equal steps in $x$) from a nonlinear one at a glance.

How the SOL examines this topic

• Fill-in-the-blank. Compute the slope from two points or a table and type it.
• Multiple choice. Interpret the slope or intercept in a context, or identify them from an equation.
• Coordinate-plane items. Read slope from a graph or place a line with a given slope.

Try this

Q1. Find the slope through $(2, 1)$ and $(6, 9)$. [1 point]

• Cue. $\frac{9 - 1}{6 - 2} = \frac{8}{4} = 2$.

Q2. In $y = -3x + 7$, what is the rate of change? [1 point]

• Cue. The slope $-3$ (output drops $3$ per unit of input).