1.1

Basic operation

\$"a" + "b" = "b" + "a"\$

Addition

1.2

Basic operation

\$"a" − "b" = −("b" − "a")\$

Subtraction

1.3

Basic operation

\$"a" times "b" = "b" times "a"\$

Multiplication

1.4

Basic operation

\$a/b = b/a\$ (for b≠0)

Division

2

Distributive Property

\$"a" times ("b" + "c") = "a" times "b" + "a" times "c"\$

3.1

Exponents

\$"a"^"m" times "a"^"n" = "a"^("m" + "n")\$

Product Rule

3.2

Exponents

\$"a"^"m" / "a"^"n" = "a"^("m" − "n")\$

Quotient Rule

3.3

Exponents

\$("a"^("m"))^("n") = "a"^("mn")\$

Power Rule

3.4

Exponents

\$"a"^(−"n") = 1/"a"^("n")\$

Negative Exponent Rule​

3.5

Exponents

\$"a"^0 = 1\$

Zero Exponent Rule

4.1

Factoring

\$"a"^2 − "b"^2 = ("a" + "b")("a" − "b")\$

Difference of Squares

4.2

Factoring

\$("a" + "b")^2 = "a"^2 + 2"ab" + "b"^2\$

Perfect Square Trinomial

4.3

Factoring

\$("a" - "b")^2 = "a"^2 -2"ab" + "b"^2\$

Perfect Square Trinomial

5.1

Factoring

\$("a" + "b")^3 = "a"^3 + 3"a"^2"b" + 3"ab"^2 + "b"^3\$

Cubic trinomial

5.2

Factoring

\$("a" - "b")^3 = "a"^3 - 3"a"^2"b" + 3"ab"^2 + "b"^3\$

Cubic trinomial

6

Factoring

Solve \$"ax"^2+ "bx" + "c" = 0 (a ≠ 0)\$

\$"x" = (-"b" ± \sqrt(b^(2) - 4"ac"))/ (2"a")\$

If \$"b"^2- 4"ac" > 0\$ - Two real unequal solution

If \$"b"^2 - 4"ac" = 0\$ - Repeated real solution

If \$"b"^2 - 4"ac" < 0\$ - Two complex solutions

7

Distance Formula

If \$"p"_1 = ("x"_1 , "y"_1)\$ and \$"p"_2 = ("x"_2, "y"_2)\$ are two points the distance between them is
\$("p"_1, "p"_2) = \sqrt{("x"_2 - "x"_1)^2 + ("y"_2 - "y"_1)^2}\$

8

Properties of Inequalities

If a < b then a + c < b + c and a - c < b - c
If a < b then c > 0 then ac < bc and \$"a"/"c" < "b"/"c"\$
If a < b then c < 0 then ac > bc and \$"a"/"c" > "b"/"c"\$

9

Absolute Value equations/Inequalities

\$|P| = "b" = "p" = -"b" or "p" = "b"\$
\$|P| < "b" = -"b" < "p" < "b"\$
\$|P| > "b" = "p" < -"b" or "p" > "b"\$

10

Complex Numbers

\$"i" = \sqrt(-1)\$ \$"i"^2 = -1\$ \$\sqrt("a") = "i"\sqrt("a")\$ a ≥ 0

(a + bi) + ( c + di) = a + c + (b + d)i

(a + bi) - ( c + di) = a - c + (b - d)i

(a + bi) (c + di) = ac - bd + (ad + bc)i

\$("a" + "bi") ("a" - "bi") = "a"^2 + "b"^2\$

\$|"a" + "bi"| = \sqrt("a"^2 + "b"^2)\$ Complex Modulus

\$\overline("a" + "bi") = "a" - "bi"\$ Complex Conjugate

\$(\overline(("a" + "bi") ("a" + "bi"))) = |"a" + "bi"|^2\$

11

Binomial Theorem

For positive integers n and any real numbers a and b:
\$("a" + "b")^n = ∑_("n" to "k" = 0) \binom{"n"}{"k"} "a"^("n" - "k") "b"^"k"\$

12

Properties of Radicals

\$root("n")("a") = "a"^(1/"n")\$

\$root("m")("n") ("a") = root("nm") ("a")\$

\$root("n") ("a"^"n") = "a"\$ if n is odd

\$root("n") ("a"^"n") = |"a"|\$ if n is even

\$root("n")("ab") = root("n") ("a") root("n") ("b")\$

\$root("n") ("a"/"b") = root("n") ("a") / root("n") ("b")\$

13

Function formulas

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

(αf)(x) = αf(x)

\$("fg")("x") = "f"("x") times "g"("x") \$

\$ ("f"/"g")("x") = "f"("x")/"g"("x") \$

\$("gof")("x") = "g"("f"("x"))\$

\$"ho"("gof")("x") = ("hog")"of"("x")\$

\$("gof")^(-1)("x") = "f"^(-1)"og"^(-1)("x") \$

14

Linear function

f(x) = mx + b

where m is the slope of the line and b is the y-intercept

15

Quadratic Function

\$ "f"("x") = "ax"^2 + "bx" + "c" \$

where a, b, and c are constants and a ≠ 0.

16

Cubic Function

\$ "f"("x") = "ax"^3 + "bx"^2 + "cx" + "d" \$

where a, b, c, and d are constants and a ≠ 0.

17

Exponential Function

\$ "f"("x") = "a" times "b"^"x" \$

where a and b are constants, and b is the base of the exponential.

18

Logarithmic Function

\$ "f"("x") = log_"b" ("x") \$

where b is the base of the logarithm.

19

Trigonometric Functions

f(x) = sin (x)

f(x) = cos (x)

f(x) = tan (x)

f(x) = csc (x)

f(x) = sec (x)

f(x) = cot (x)

Sine Function

Cosine Function

Tangent Function

Cosecant Function

Secant Function

Cotangent Function

20

Square Root Function

\$ "f"("x") = \sqrt "x" \$

21

Absolute Value Function

\$ "f"("x") = | "x" | \$