Fiveable SAT Math: Formulas to Memorize

SAT Math: Essential Formulas To Memorize

The SAT math section provides a reference sheet, but it doesn't cover everything you'll need. Several key formulas show up repeatedly on the test and aren't on that sheet. This guide covers the formulas worth memorizing so you're not stuck trying to derive them under time pressure.

SAT Math: Linear Line Formulas

Linear equations show up constantly on the SAT. They can be written in three main forms, and knowing when to use each one will save you time.

The Standard Form of Linear Equations

$Ax + By = C$

• A: coefficient of x
• B: coefficient of y
• x and y: variables
• C: constant

Standard form is useful when a problem gives you information about both x and y together, or when you need to find intercepts quickly. To find the x-intercept, set $y = 0$. To find the y-intercept, set $x = 0$.

The Slope-Intercept Form of Linear Equations

$y = mx + b$

• m: slope of the line
• b: y-intercept (where the line crosses the y-axis)

This is the form you'll use most often. The slope (m) tells you how steep the line is, and the y-intercept (b) tells you where the line crosses the y-axis. If a problem asks you to identify the slope or y-intercept, rearrange the equation into this form.

The Point-Slope Form of Linear Equations

$y - y_1 = m(x - x_1)$

• $(x_1, y_1)$: coordinates of a known point on the line
• m: slope

Use this form when you're given a slope and a specific point (or two points, since you can calculate the slope first). It's the fastest way to write an equation without needing to solve for b.

The Slope of Linear Lines

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

• $(x_1, y_1)$: coordinates of the first point
• $(x_2, y_2)$: coordinates of the second point

This is the "rise over run" formula. It doesn't matter which point you call "first" or "second," but make sure $x_1$ and $y_1$ come from the same point.

The Midpoint Formula

$\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

This finds the exact center between two points. You're just averaging the x-coordinates and averaging the y-coordinates separately. Your answer should be written as an (x, y) coordinate pair.

The Distance Formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This calculates the straight-line distance between two points on a coordinate plane. It comes directly from the Pythagorean theorem: the horizontal and vertical differences form the two legs of a right triangle, and the distance is the hypotenuse.

SAT Math: Distance/Rate Formula

$\text{Distance} = \text{Speed} \times \text{Time}$

• Distance: total distance traveled
• Speed: rate of motion (also called rate)
• Time: duration of travel

You can rearrange this to solve for any variable: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ or $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$. This formula applies to objects moving at a constant speed.

SAT Math: Quadratic Equations/Parabolas

Quadratic equations describe parabolas. Like linear equations, they come in multiple forms, and each form reveals different information.

Standard Form of a Quadratic Equation

$ax^2 + bx + c = 0$

• a, b, and c: constants (where $a \neq 0$)
• If a is positive, the parabola opens upward
• If a is negative, the parabola opens downward
• c is the y-intercept (the value when $x = 0$)

The Vertex Form of a Quadratic Equation

$y = a(x - h)^2 + k$

• a: determines the direction and width of the parabola
• (h, k): the vertex of the parabola

The vertex is the highest or lowest point on the parabola. Watch the sign: in $y = a(x - 3)^2 + 5$, the vertex is $(3, 5)$, not $(-3, 5)$.

The Factored Form of a Quadratic Equation

$y = a(x - r_1)(x - r_2)$

• a: coefficient of the quadratic term
• $r_1$ and $r_2$: the roots (also called zeros or x-intercepts)

The roots are the x-values where the parabola crosses the x-axis. Setting each factor equal to zero gives you the solutions: $x = r_1$ and $x = r_2$. This form is the fastest way to identify the x-intercepts.

Coordinates of the Vertex from Standard Form

When you have a quadratic in standard form and need the vertex:

1. Find the x-coordinate: $x = \frac{-b}{2a}$
2. Find the y-coordinate: plug that x-value back into the original equation

The x-coordinate also gives you the axis of symmetry of the parabola.

SAT Math: Circle Formulas

Arc Length Formula: $L = 2\pi r \left(\frac{\theta}{360}\right)$

• L: arc length
• r: radius
• θ: central angle in degrees

Sector Area Formula: $A = \pi r^2 \left(\frac{\theta}{360}\right)$

• A: sector area
• r: radius
• θ: central angle in degrees

Center-Radius Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$

• (h, k): center of the circle
• r: radius

SAT Math: Exponents/Roots Formulas

These rules come up frequently when simplifying expressions.

Product of Powers: $a^m \cdot a^n = a^{m+n}$

Power of a Power: $(a^m)^n = a^{mn}$

Power of a Product: $(ab)^n = a^n \cdot b^n$

Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

Negative Exponent: $a^{-n} = \frac{1}{a^n}$

SAT Math: Exponential Function/Compound Interest

General form of an exponential function: $f(x) = a \cdot b^x$

• a: initial value (y-intercept, the value when $x = 0$)
• b: the base, which determines growth or decay
  • If $b > 1$: exponential growth
  • If $0 < b < 1$: exponential decay

Compound Interest Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• A: total amount after interest
• P: principal (starting amount)
• r: annual interest rate (as a decimal, so 5% = 0.05)
• n: number of times interest compounds per year
• t: number of years

Continuous Compound Interest Formula: $A = Pe^{rt}$

• e: Euler's number (approximately 2.71828)
• All other variables are the same as above

Use this version when a problem specifies "continuous" compounding. Instead of compounding at intervals (monthly, quarterly), the interest compounds constantly.

SAT Math: Trigonometry Functions

Image Courtesy of MathsisFun.

The mnemonic SOH-CAH-TOA is the fastest way to remember the three main trig ratios for a right triangle:

Sine (sin): $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Cosine (cos): $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Tangent (tan): $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

The reciprocal functions flip these ratios:

Cosecant (csc): $\csc(\theta) = \frac{1}{\sin(\theta)}$

Secant (sec): $\sec(\theta) = \frac{1}{\cos(\theta)}$

Cotangent (cot): $\cot(\theta) = \frac{1}{\tan(\theta)}$

For the SAT, sin, cos, and tan are by far the most tested. The reciprocal functions rarely appear, but they're quick to derive if you know the first three.