COLLEGE ALGEBRA
COMPREHENSIVE STUDY GUIDE
The Ultimate High-Yield Reference for Exams, Formulas, & Graphs — 2026 Edition
1. Quadratic Equations & Complex Numbers 3. Exponential & Logarithmic Functions
Standard Form: ax2 + bx + c = 0 Properties of Logarithms (b > 0, b ̸= 1):
• Quadratic Formula: logb (xy) = logb (x) + logb (y) (Product)
√
x
−b ± b2 − 4ac logb = logb (x) − logb (y) (Quotient)
x= y
2a
logb (xr ) = r logb (x) (Power)
• The Discriminant (D = b − 4ac): 2
ln(a)
logb (a) = (Change of Base)
– D > 0 ⇒ 2 distinct real roots. ln(b)
– D = 0 ⇒ 1 repeated real root. Symmetry of Inverse Functions: An exponential
– D < 0 ⇒ 2 complex conjugate roots. function and its corresponding logarithmic function
√ are inverses of each other, reflecting perfectly across
• Complex Numbers: i = −1, i2 = −1. the line y = x.
Division rule: Multiply numerator and denomi-
nator by the complex conjugate of the denomi- y
nator (a − bi). 2x
4
log2 (x)
y=x
2
2. Functions & Graph Transforma-
x
tions −2 2 4
Finding the Domain −2
1. Denominators: Set denominator ̸= 0.
2. Even Roots: Set the radicand (inside of root) ≥ 0. 4. System of Linear & Non-Linear
3. Logarithms: Set argument > 0. Equations
For a system of equations, the solution represents the phys-
Transformation Rules & Parabolic Shifts ical point of intersection between the lines or curves.
Let f (x) = x2 be the parent function. y
4
• Vertical Shift Up c units: f (x) + c
2 Intersection (1, 2)
• Horizontal Shift Right c units: f (x − c) x
−1 1 2 3 4
6 y
x2
(x −42)2 + 1
Cramer’s Rule for 2 Variables
2 For the system: (
x a1 x + b1 y = c1
−2 2 4
a2 x + b2 y = c2
Dx Dy
Symmetry & Function Types x= , y= (D ̸= 0)
D D
• Even Function (y-axis symmetry): f (−x) = f (x) Where:
• Odd Function (Origin symmetry): f (−x) = −f (x) a1 b1 c1 b1 a1 c1
D= , Dx = , Dy =
a2 b2 c2 b2 a2 c2
COMPREHENSIVE STUDY GUIDE
The Ultimate High-Yield Reference for Exams, Formulas, & Graphs — 2026 Edition
1. Quadratic Equations & Complex Numbers 3. Exponential & Logarithmic Functions
Standard Form: ax2 + bx + c = 0 Properties of Logarithms (b > 0, b ̸= 1):
• Quadratic Formula: logb (xy) = logb (x) + logb (y) (Product)
√
x
−b ± b2 − 4ac logb = logb (x) − logb (y) (Quotient)
x= y
2a
logb (xr ) = r logb (x) (Power)
• The Discriminant (D = b − 4ac): 2
ln(a)
logb (a) = (Change of Base)
– D > 0 ⇒ 2 distinct real roots. ln(b)
– D = 0 ⇒ 1 repeated real root. Symmetry of Inverse Functions: An exponential
– D < 0 ⇒ 2 complex conjugate roots. function and its corresponding logarithmic function
√ are inverses of each other, reflecting perfectly across
• Complex Numbers: i = −1, i2 = −1. the line y = x.
Division rule: Multiply numerator and denomi-
nator by the complex conjugate of the denomi- y
nator (a − bi). 2x
4
log2 (x)
y=x
2
2. Functions & Graph Transforma-
x
tions −2 2 4
Finding the Domain −2
1. Denominators: Set denominator ̸= 0.
2. Even Roots: Set the radicand (inside of root) ≥ 0. 4. System of Linear & Non-Linear
3. Logarithms: Set argument > 0. Equations
For a system of equations, the solution represents the phys-
Transformation Rules & Parabolic Shifts ical point of intersection between the lines or curves.
Let f (x) = x2 be the parent function. y
4
• Vertical Shift Up c units: f (x) + c
2 Intersection (1, 2)
• Horizontal Shift Right c units: f (x − c) x
−1 1 2 3 4
6 y
x2
(x −42)2 + 1
Cramer’s Rule for 2 Variables
2 For the system: (
x a1 x + b1 y = c1
−2 2 4
a2 x + b2 y = c2
Dx Dy
Symmetry & Function Types x= , y= (D ̸= 0)
D D
• Even Function (y-axis symmetry): f (−x) = f (x) Where:
• Odd Function (Origin symmetry): f (−x) = −f (x) a1 b1 c1 b1 a1 c1
D= , Dx = , Dy =
a2 b2 c2 b2 a2 c2
