Linear Equations

y = mx + b

Slope-intercept form. m is slope and b is y-intercept.

Line: y = 4x + 7. What is the y-intercept?

Answer: 7

m = y₂ - y₁x₂ - x₁

Finds slope between two points.

(2,3) and (6,11). Find slope.

Answer: 2

Quadratics

x = -b ± √(b² - 4ac)2a

Used to solve quadratic equations.

Solve x² − 5x + 6 = 0

Answer: x = 2 and x = 3

b² − 4ac

Discriminant determines the number of real solutions.

b² - 4ac < 0 → No real solutions
b² - 4ac = 0 → One real solution
b² - 4ac > 0 → Two real solutions

For x² − 6x + 9 = 0

Answer: 0

Vertex: x = -b2a

Finds the x-coordinate of a parabola’s vertex.

f(x) = x² − 4x + 3

Answer: x = 2, y = -1 → Vertex (2, -1)

Sum of Solutions = -b/a

For ax² + bx + c = 0, the sum of all solutions is -b/a.

x² - 5x + 6 = 0

Answer: 5

Product of Solutions = c/a

For ax² + bx + c = 0, the product of all solutions is c/a.

x² - 5x + 6 = 0

Answer: 6

f(a)=0 ⇒ (x-a) is a factor

Factor Theorem. If f(a)=0, then (x-a) is a factor of the polynomial.

f(x)=x²-9. Is (x-3) a factor?

Answer: Yes, because f(3)=0

Triangles

Area = 12bh

Area of a triangle.

Base 8, height 6.

Answer: 24

a² + b² = c²

Pythagorean theorem.

Sides 5 and 12.

Answer: 13

a - b < c < a + b

Triangle Inequality Theorem. c is the unknown side length and a ≥ b.

Sides 8 and 5. Possible values of c?

Answer: 3 < c < 13

Coordinate Geometry

M = (x₁ + x₂, y₁ + y₂) 2

Finds the point exactly halfway between two coordinates.

(2, 4) and (8, 10)

Answer: (5, 7)

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Finds the distance between two points on a coordinate plane.

(1, 2) and (4, 6)

Answer: 5

3D Shapes (Volumes)

V = lwh

Volume of a rectangular prism (box).

l = 4, w = 3, h = 2

Answer: 24

V = πr²h

Volume of a cylinder.

r = 3, h = 5

Answer: 45π

V = 13πr²h

Volume of a cone.

r = 2, h = 9

Answer: 12π

V = 43πr³

Volume of a sphere.

r = 3

Answer: 36π

V = 13bh

Volume of a pyramid. b = area of base.

Base area b = 18, h = 6

Answer: 36

Circles

Area = πr²

Area inside a circle.

Radius 4.

Answer: 16π

C = 2πr

Circumference of a circle.

Radius 5.

Answer: 10π

(x - h)² + (y - k)² = r²

Standard equation of a circle.

Center (2, -3), radius = 4

Answer: (x - 2)² + (y + 3)² = 16

A = ½θr²

Area of a sector when θ is measured in radians.

θ = π/2, r = 4

Answer: 4π

A = (θ/360)(πr²)

Area of a sector when θ is measured in degrees.

θ = 90°, r = 6

Answer: 9π

s = θr

Arc length formula. θ must be in radians.

θ = π/2, r = 8

Answer: 4π

Trigonometry

sin θ = opphyp

Sine ratio.

Opp = 3, Hyp = 5

Answer: 35

45-45-90 → 1:1:√2

Find Hypotenuse

Short Leg = 8

Answer: 8√2

30-60-90 → 1:√3:2

Find Long Leg and Hypotenuse

Short Leg = 5

Answer: Long = 5√3, H = 10

cos(A)=sin(B), sin(A)=cos(B)

If A + B = 90°, then these complementary angle identities apply.

A = 30°, B = 60°

Answer: cos(30°)=sin(60°)

Statistics

Mean = sumcount

Average value.

2, 4, 6, 8

Answer: 5

P(A) = favorabletotal

Basic probability.

Roll a die. Probability of 3?

Answer: 16

% Change = (new - old)old × 100

Measures increase/decrease.

100 → 125

Answer: 25%

Good To Know

Exponential Function

y = a(b)^x

a = y-intercept (initial value)
b = growth/decay factor
b > 1 = growth
0 < b < 1 = decay

Shape of a Data Distribution

Mean vs. Median

Left-Skewed → Mean < Median
Symmetric → Mean = Median
Right-Skewed → Mean > Median

Standard Form

Ax + By = C

Slope = -A/B
Y-Intercept = C/B

Extraneous Solutions

Always Check Your Answers

Radical equations, rational equations, and some SAT problems can create solutions that do not actually work.

Plug answers back into the original equation.

Linear Equations

ax+b = cx+d

Same x-coefficient but different constants → No Solution

Different x-coefficients → One Solution

Parallel & Perpendicular Lines

Parallel: same slope (m₁ = m₂)
Perpendicular: negative reciprocals (m₁ · m₂ = -1)

Central & Inscribed Angles

Central Angle Theorem

A central angle is always twice the measure of an inscribed angle that intercepts the same arc.

Inscribed Angle = θ
Central Angle = 2θ