Find slope and write linear functions in slope-intercept and point-slope form from a graph, a description, or two points (NC.M1.F-LE.2, F-BF.1a).

What this topic is asking

NC.M1.F-LE.2 asks you to construct linear functions from a graph, a description, or two input-output pairs, and NC.M1.F-BF.1a asks you to build a function rule from those same kinds of information. The hinge is slope: once you have the slope and one point, you can write the line. This is among the highest-frequency skills on the EOC.

Slope is a rate of change

Slope measures how steeply and in which direction a line rises.

In a context, slope is the per-unit rate: dollars per hour, miles per gallon, growth per year.

The two forms for writing a line

Both forms are essential because the EOC gives information in different ways. Since NC Math 1 has no reference sheet, memorize both.

• Slope-intercept form $y = mx + b$: best when you know the slope $m$ and the y-intercept $b$ (the value when $x = 0$).
• Point-slope form $y - y_1 = m(x - x_1)$: best when you know the slope and any point $(x_1, y_1)$.

Writing a line from two points

Writing a line from a context

In a real situation, read the slope as the rate and the intercept as the starting value. "A pool starts with $200$ gallons and fills at $15$ gallons per minute" gives $y = 15x + 200$, where $15$ is the slope (rate) and $200$ is the y-intercept (initial amount). This is the same structure as creating equations.

How the NC Math 1 EOC examines this topic

• Multiple choice. Choose the equation of a line from a graph, two points, or a description.
• Gridded response. Enter a slope, an intercept, or a value computed from the line.
• Technology-enhanced. Plot a line, or match equations to graphs.

Slope as a rate of change connects directly to average rate of change for functions in general and to the line of best fit in statistics, where slope and intercept are interpreted in context.

Why slope plus a point fixes a line

A line is completely determined by two facts: how steep it is (slope) and where it sits (one point). Given only a slope, infinitely many parallel lines qualify; given only a point, infinitely many lines pass through it; together they pin down exactly one. That is why every line-writing problem reduces to "find the slope, then anchor with a point." Point-slope form is the most general tool because it accepts any point, not just the intercept, so it always works even when the line never crosses a convenient grid mark. Internalizing this, two facts determine a line, makes the whole topic a single repeatable move.

Try this

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

• Cue. $m = \dfrac{9 - 1}{2 - (-2)} = \dfrac{8}{4} = 2$.

Q2. Write the line with slope $4$ through $(3, 5)$. [2 points]

• Cue. $y - 5 = 4(x - 3) \Rightarrow y = 4x - 7$.