Determine the slope and intercepts of a linear function, write its equation in slope-intercept, point-slope, and standard form, and graph it, including parallel and perpendicular lines (MA.912.AR.2.3, MA.912.AR.3.1).

What this topic is asking

MA.912.AR.3 is about linear functions: finding the slope and intercepts, writing the equation in slope-intercept, point-slope, and standard form, graphing the line, and using parallel and perpendicular slope relationships. All the forms and the slope formula are on the reference sheet, so the B.E.S.T. Algebra 1 EOC tests applying them quickly and correctly.

Slope and intercepts

The slope measures steepness as rise over run. From two points:

The $y$-intercept is where the line crosses the $y$-axis ($x = 0$); the $x$-intercept is where it crosses the $x$-axis ($y = 0$). In slope-intercept form $y = mx + b$, the slope is $m$ and the $y$-intercept is $b$.

The three forms (reference sheet)

Use slope-intercept to graph or read features, point-slope to write an equation from a slope and a point, and standard form when the problem or answer key calls for it.

Graphing

Plot the $y$-intercept $(0, b)$, then use the slope as rise over run to step to a second point (for $m = \frac{2}{3}$, go up 2 and right 3). Connect them. A positive slope rises left to right; a negative slope falls; a zero slope is horizontal; an undefined slope is vertical.

Parallel and perpendicular

• Parallel lines never meet: they have the same slope, different intercepts.
• Perpendicular lines meet at a right angle: their slopes are opposite reciprocals, so $m_1 \cdot m_2 = -1$. A slope of $\frac{2}{5}$ pairs with $-\frac{5}{2}$.

How the B.E.S.T. EOC examines this topic

• Equation editor. Write the equation of a line from points, a slope and a point, or a description.
• GRID and matching. Graph a line, or match equations to graphs.
• Multiple choice. Identify slope, intercepts, or a parallel or perpendicular slope.

A clarifying idea: the three forms are the same line wearing different clothes, chosen for what the task needs. Slope-intercept exposes the slope and start; point-slope is fastest to write from raw data; standard form is tidy for systems. Converting between them is just algebra.

Why perpendicular slopes are opposite reciprocals

The opposite-reciprocal rule encodes a right angle. A slope of $\frac{2}{5}$ means "right 5, up 2." Rotating that direction by $90^\circ$ turns "right 5, up 2" into "up 5, left 2," which as rise over run is $\frac{5}{-2} = -\frac{5}{2}$, the opposite reciprocal. The two steps in the numerator and denominator swap (reciprocal) and one sign changes (opposite) precisely because a quarter turn exchanges the horizontal and vertical movements and reverses one of them. The product being $-1$ is the algebraic fingerprint of that perpendicularity, and checking the product is the quickest way to confirm two lines meet at a right angle on the EOC.

Interpreting slope and intercept in context

On modeling items, the slope and $y$-intercept carry real meaning, and the EOC awards credit for stating it. For a phone plan modeled by $C = 0.10m + 25$, the slope $0.10$ is the rate, the cost added per minute, and the $y$-intercept $25$ is the starting value, the fixed monthly fee charged at zero minutes. For a tank draining by $h(t) = -3t + 48$, the slope $-3$ means the water level falls 3 units per minute (negative because it is decreasing), and the intercept $48$ is the initial level. The sign of the slope tells you increasing versus decreasing, and its size tells you how fast. A reliable habit is to read the slope as "output units per one input unit" and the intercept as "the output when the input is zero," then phrase the answer in the situation's own words (dollars, gallons, minutes), which is exactly what earns the interpretation points.

Try this

Q1. Find the slope between $(-2, 4)$ and $(2, -4)$. [1 point]

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

Q2. What slope is parallel to $y = -3x + 7$? [1 point]

• Cue. The same slope, $-3$.