Reference Sheet · SAT Math · Algebra
Linear Functions & Lines
Slope, line equations, and the coordinate-geometry formulas that go with them. Each card includes a brief worked example.
Slope
Slope formula
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
m = slope
(x₁, y₁), (x₂, y₂) = any two points on the line
(x₁, y₁), (x₂, y₂) = any two points on the line
"Rise over run." Pick either point as point 1 — but be consistent (don't swap halfway).
EXAMPLE
Find the slope of the line through $(2, 5)$ and $(6, 13)$.
$m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = $ $2$
Special slopes
$$\text{horizontal: } m = 0$$$$\text{vertical: } m \text{ is undefined}$$
A horizontal line has zero rise → slope $0$. A vertical line has zero run → division by zero, undefined.
EXAMPLE
What's the slope of the line $y = 4$?
It's horizontal (no $x$ term, $y$ is fixed): $m = 0$
Three forms of a line
Slope-intercept form
$$y = mx + b$$
m = slope
b = y-intercept (where the line crosses the y-axis)
b = y-intercept (where the line crosses the y-axis)
The most common form on the SAT. Use when you know the slope and y-intercept, or when you need to graph quickly.
EXAMPLE
Write the equation of the line with slope $3$ and y-intercept $-4$.
$y = 3x - 4$
Try it yourself
- Find the sliders for $m$ (slope) and $b$ (y-intercept) in the panel on the left.
- Drag $m$ to make the line steeper or shallower. Try negative values to flip the direction.
- Drag $b$ up and down. Watch where the line crosses the y-axis.
Notice: $m$ controls the tilt of the line; $b$ controls where it crosses the y-axis. Setting $m = 0$ gives a horizontal line; making $m$ very large makes it nearly vertical.
Point-slope form
$$y - y_1 = m(x - x_1)$$
m = slope
(x₁, y₁) = a known point on the line
(x₁, y₁) = a known point on the line
Use when you know the slope and one point. Often faster than slope-intercept because you don't have to solve for $b$ separately.
EXAMPLE
Write the equation of the line with slope $2$ passing through $(3, 7)$.
$y - 7 = 2(x - 3)$
Solving for $y$ gives $y = 2x + 1$ in slope-intercept form.
Standard form
$$Ax + By = C$$
A, B, C = constants (typically integers)
Less common on the SAT, but appears in word problems where each variable represents a quantity. Slope from standard form: $m = -A/B$.
EXAMPLE
Find the slope of $3x + 4y = 12$.
$m = -\dfrac{A}{B} = -\dfrac{3}{4} = $ $-\dfrac{3}{4}$
Coordinate-geometry formulas
Distance formula
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
d = distance between two points
(x₁, y₁), (x₂, y₂) = the two points
(x₁, y₁), (x₂, y₂) = the two points
This is just the Pythagorean theorem in disguise — the horizontal and vertical legs of a right triangle, with the distance as the hypotenuse.
EXAMPLE
Find the distance between $(1, 2)$ and $(4, 6)$.
$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = $ $5$
Midpoint formula
$$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$$
M = midpoint of the segment
(x₁, y₁), (x₂, y₂) = endpoints
(x₁, y₁), (x₂, y₂) = endpoints
Just the average of the x-coordinates and the average of the y-coordinates.
EXAMPLE
Find the midpoint of $(2, -3)$ and $(8, 5)$.
$M = \left(\dfrac{2+8}{2}, \dfrac{-3+5}{2}\right) = $ $(5, 1)$
Parallel & perpendicular
Parallel lines
$$m_1 = m_2$$
Parallel lines have equal slopes and never intersect.
EXAMPLE
Find a line parallel to $y = 3x + 2$ passing through $(0, 7)$.
Same slope $3$, new y-intercept $7$:
$y = 3x + 7$
Perpendicular lines
$$m_1 \cdot m_2 = -1$$
Perpendicular slopes are negative reciprocals of each other. Flip the fraction, change the sign.
EXAMPLE
Find a line perpendicular to $y = \dfrac{2}{3}x + 1$ passing through $(0, 4)$.
Negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$:
$y = -\dfrac{3}{2}x + 4$
Try it yourself
- Find the slider for $m$ and drag it.
- Watch all three lines change at once.
- The red and blue lines stay parallel — same slope, different y-intercepts.
- The green line stays perpendicular to them.
Notice: When you change $m$, the parallel line's slope matches red's exactly. The perpendicular line's slope is the negative reciprocal of $m$ — flip the fraction, change the sign.