Linear Functions and Slope

What Slope Really Measures

On SAT Math, a linear function is any relationship whose graph is a straight line and whose rate of change is constant. The one idea the test returns to again and again is slope — the number that says how fast the output changes as the input increases. For two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line,

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Read it as "rise over run": how far $y$ climbs (or falls) for each one-unit increase in $x$. A positive slope rises left to right, a negative slope falls, a slope of $0$ is a horizontal line, and a vertical line has an undefined slope because the run in the denominator is zero.

Slope as a Rate of Change

The SAT tests slope far more often as an interpretation than as a bare calculation. If a phone plan costs $C = 15 + 0.10t$ dollars for $t$ text messages, the slope $0.10$ is not just a number — it is "10 cents per text." Whenever a coefficient multiplies the input variable, ask yourself "per what?" That phrase almost always names the correct answer on interpretation questions. The constant term $15$, by contrast, is the starting value — the cost when $t = 0$.

The Three Forms You Must Recognize

A line can be written three ways, and the SAT expects you to switch between them without hesitation.

Form	Looks like	Best for
Slope-intercept	$y = mx + b$	Reading slope $m$ and $y$-intercept $b$ instantly
Standard	$Ax + By = C$	Finding intercepts quickly; setting up systems
Point-slope	$y - y_1 = m(x - x_1)$	Building a line from one point and a slope

To convert standard form to slope-intercept, solve for $y$. Starting from $3x + 2y = 12$:

$\begin{aligned} 3x + 2y &= 12 \\ 2y &= -3x + 12 \\ y &= -\tfrac{3}{2}x + 6 \end{aligned}$

Now the slope $-\tfrac{3}{2}$ and the $y$-intercept $6$ are visible at a glance.

Building a Line from Two Points

Suppose a line passes through $(2, 5)$ and $(6, 13)$. First compute the slope:

$m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$

Then feed the slope and one point into point-slope form and simplify:

$\begin{aligned} y - 5 &= 2(x - 2) \\ y - 5 &= 2x - 4 \\ y &= 2x + 1 \end{aligned}$

The equation is $y = 2x + 1$. Always confirm with the other point: $y = 2(6) + 1 = 13$. It matches, so you are done. The slope $2$ means $y$ rises by 2 for each 1-unit increase in $x$; the intercept $1$ is the value of $y$ when $x = 0$.

Parallel and Perpendicular Slopes

Two relationships between lines drive most coordinate-geometry questions:

• Parallel lines have the same slope (and different $y$-intercepts, or they would be the same line). They never intersect.
• Perpendicular lines have slopes that are negative reciprocals — their product is $-1$. If one line has slope $\tfrac{3}{4}$, a line perpendicular to it has slope $-\tfrac{4}{3}$.

The perpendicular rule has two moves, and you must do both: flip the fraction and change the sign. Flipping without switching the sign — turning $\tfrac{3}{4}$ into $\tfrac{4}{3}$ — is the single most common perpendicular-slope mistake.

Finding a Missing Coordinate

The SAT sometimes gives you the slope and one full point, then hides a coordinate of the second point. Use the slope formula as an equation. If the line through $(a, 3)$ and $(5, 11)$ has slope $2$:

$\begin{aligned} \frac{11 - 3}{5 - a} &= 2 \\ \frac{8}{5 - a} &= 2 \\ 8 &= 2(5 - a) \\ 8 &= 10 - 2a \\ a &= 1 \end{aligned}$

The missing coordinate is $a = 1$. This "solve for the coordinate" pattern is just the slope formula turned into an ordinary linear equation — which is exactly why solving linear equations is a prerequisite here.

The Habits That Pay Off

Keep the subtraction order consistent between numerator and denominator, never confuse the constant $b$ with the slope $m$, and remember that horizontal ($m = 0$) and vertical (undefined) lines are opposites, not the same. Master those and slope questions become some of the fastest points on the section.