Find the slope of a line from two points, write linear equations in slope-intercept and point-slope form, and interpret slope as a constant rate of change in context.

What this topic is asking

The Functions category requires confident work with linear functions (the F-IF, F-LE, and F-BF standards). On the Grade 10 MCAS you compute slope from two points, write a line's equation in slope-intercept and point-slope form, and interpret slope as a constant rate of change in a context. Linear functions are the most frequently tested function type, and because the slope formula is not on the reference sheet, you must know it cold.

Finding slope

Slope is the constant ratio of vertical change to horizontal change between any two points on the line:

For $(2, 3)$ and $(6, 11)$, $m = \dfrac{11 - 3}{6 - 2} = \dfrac{8}{4} = 2$. The one rule that prevents errors: subtract the coordinates in the same order in the top and bottom. Subtracting $y$ as "second minus first" but $x$ as "first minus second" flips the sign.

The sign and size of slope describe the line: positive rises left to right, negative falls, a slope of 0 is horizontal (a constant function), and a vertical line has undefined slope because the run is zero (division by zero).

Writing the equation of a line

Two forms cover almost every MCAS task:

• Slope-intercept form $y = mx + b$ is best when you know the slope and the y-intercept, or when a question asks you to read them off. The slope is $m$; the y-intercept is $b$ (the value when $x = 0$).
• Point-slope form $y - y_1 = m(x - x_1)$ is best when you know the slope and any point. Plug in the slope and the point's coordinates, then simplify to slope-intercept if needed.

Slope as a rate of change

In a context, slope is a constant rate of change: how much the output changes for each one-unit increase in the input. For a cost $C = 40 + 15m$, the slope 15 is "dollars per month"; for a filling pool, the slope is "gallons per minute". The y-intercept is the starting value (the output when the input is zero), such as a fixed fee.

Identifying the rate from a table means checking that equal steps in $x$ give equal steps in $y$. If they do, the function is linear and that common difference (per unit) is the slope. If the differences are not constant, the function is not linear.

Parallel and perpendicular lines

Two non-vertical lines are parallel when they have the same slope and different intercepts. They are perpendicular when their slopes are negative reciprocals, meaning the product of the slopes is $-1$. A line with slope $\frac{2}{3}$ is perpendicular to one with slope $-\frac{3}{2}$, because $\frac{2}{3} \cdot \left(-\frac{3}{2}\right) = -1$. The MCAS uses this to ask for a line parallel or perpendicular to a given one through a stated point.

Try this

Q1. Find the slope through $(-1, 4)$ and $(3, -4)$.

• Cue. $\frac{-4 - 4}{3 - (-1)} = \frac{-8}{4} = -2$.

Q2. Write the line with slope $-3$ through $(0, 5)$.

• Cue. $y = -3x + 5$.