Identify linear functions by their constant rate of change, compute average rate of change from tables and graphs, and interpret slope and intercept in context (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard studies the linear function as a function: its defining feature is a constant rate of change. The Georgia Milestones EOC asks you to recognize a linear function from a table (equal output steps for equal input steps), to compute the rate of change from a graph or table, and to interpret the slope and intercept in a context. This is the function-domain companion to the equation-writing work in the linear module, and the two reinforce each other: the rate of change is the slope, and the starting value is the y-intercept.

The constant-rate signature

A linear function changes by a fixed amount for each unit increase in the input. In a table with equally spaced inputs, the outputs go up (or down) by the same step:

Each step adds 3, so the rate of change is 3 and the function is linear: $f(x) = 3x + 4$. If the steps were not equal, the function would not be linear.

Computing rate of change

The rate of change between two points is the change in output divided by the change in input.

From a graph, pick two clear lattice points and form the ratio. From a table, divide the output change by the input change between two rows.

Interpreting slope and intercept

In a linear model, name each parameter in the units of the situation:

• Slope is the rate: dollars per mile, inches per week, dollars per month.
• y-intercept is the starting value: the fixed fee, the initial height, the value at time 0.

A negative slope models a decrease (a balance shrinking, a tank draining); the y-intercept is still the starting value.

How the Milestones examines this topic

• Multiple choice. Find the rate of change from a table or graph, or identify a function as linear.
• Numeric entry. Compute the average rate of change over an interval.
• Constructed response. Compute a rate of change and interpret it in context, naming the units.

Why constant rate forces a straight line

The reason a constant rate of change produces a straight line is worth seeing, because it ties the table, the graph, and the equation together. If every unit step in $x$ adds the same amount $m$ to $y$, then starting from the y-intercept $b$ the outputs are $b$, $b + m$, $b + 2m$, $b + 3m$, and so on, which is exactly $y = mx + b$. Plotting those equally spaced, equally rising points gives a straight line by construction. This is also why the average rate of change is the same over every interval for a linear function: the rise per unit run never varies. Recognizing that "constant rate," "straight line," and "$y = mx + b$" are three descriptions of one object is precisely the connected understanding the Georgia standards aim for, and it lets you switch representations fluidly on the EOC.

Linear versus not linear from a table

A frequent EOC task is to decide, from a table, whether a function is linear. The test is the first differences: subtract consecutive outputs (for equally spaced inputs) and check whether they are constant. If the differences are equal, the function is linear and that common difference is the slope. If the differences themselves change, the function is not linear, and if the differences grow by a constant ratio instead, you are likely looking at an exponential function, the contrast explored in the comparing module. Computing first differences is fast and decisive, and it generalizes to spotting the function family at a glance.

Try this

Q1. A table has $f(2) = 5$, $f(4) = 11$, $f(6) = 17$. Find the rate of change. [1 point]

• Cue. $\frac{11 - 5}{4 - 2} = 3$ per unit.

Q2. In $T(h) = 60 - 4h$ (temperature after $h$ hours), interpret the slope. [1 point]

• Cue. The slope $-4$ means the temperature drops 4 degrees per hour.