Source: SYMKC Series, Calculus Prep (Texas A&M University)
Tags: algebra 1, powers, perfect squares, cubes, factorials, linear equations, quadratic equations, slope, inequalities, systems of equations, order of operations, properties, absolute value, functions, distance formula, midpoint, pythagorean theorem, central tendency
This covers the foundational algebra you need automatic recall of before moving into Algebra 2 and beyond. Perfect squares, cubes, powers of 2, linear and quadratic equation forms, slope, function definitions, and the core number properties. If you have to stop and think about any of these during a test, you are already behind.
Perfect square
The result of squaring an integer. You need instant recall of 13² through 25².
Factorial (n!)
The product of all positive integers from 1 to n. By convention, 0! = 1.
Slope
The rate of change between two points: rise over run. Calculated as (y₂ – y₁) / (x₂ – x₁).
Quadratic formula
The formula x = (–b ± √(b² – 4ac)) / 2a, used to find the roots of any quadratic equation ax² + bx + c = 0.
Discriminant
The expression b² – 4ac inside the quadratic formula. Determines the nature of the roots.
Domain
The set of all possible input values (x-values) for a function.
Range
The set of all possible output values (y-values) for a function.
Direct variation
A relationship of the form y = kx, where k is a constant. As x increases, y increases proportionally.
Indirect variation
A relationship of the form y = k/x. As x increases, y decreases.
You should be able to rattle these off without calculation:
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1,000
11³ = 1,331
12³ = 1,728
2⁴ = 16
3⁴ = 81
4⁴ = 256
5⁴ = 625
2⁵ = 32
3⁵ = 243
4⁵ = 1,024
5⁵ = 3,125
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512
2¹⁰ = 1,024
2¹¹ = 2,048
2¹² = 4,096
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
Parent function: y = x
Standard form: Ax + By = C
Slope-intercept form: y = mx + b
Point-slope form: y – y₁ = m(x – x₁)
Slope: m = (y₂ – y₁) / (x₂ – x₁)
Graph: a straight line; the parent function y = x passes through the origin at 45°
Parent function: y = x²
General form: y = ax² + bx + c
Standard form (vertex form): y = a(x – h)² + k
Intercept form (factored form): y = a(x – p)(x – q)
Vertex: (h, k) in vertex form; or (–b/2a, f(–b/2a)) from general form
Axis of symmetry: x = –b/2a
Quadratic formula: x = (–b ± √(b² – 4ac)) / 2a
Graph: a parabola opening upward if a > 0, downward if a < 0
< means "less than" (open circle on a number line)
≤ means "less than or equal to" (closed circle)
> means "greater than" (open circle)
≥ means "greater than or equal to" (closed circle)
Graphing – plot both equations and find the intersection point
Substitution – solve one equation for a variable, substitute into the other
Elimination – add or subtract equations to cancel out one variable
Parentheses (and other grouping symbols)
Exponents (and roots)
Multiplication and Division (left to right)
Addition and Subtraction (left to right)
Commutative property
Addition: a + b = b + a
Multiplication: ab = ba
Associative property
Addition: (a + b) + c = a + (b + c)
Multiplication: (ab)c = a(bc)
Distributive property
a(b + c) = ab + ac
For any real number a, the absolute value is always ≥ 0
|a| = a if a ≥ 0; |a| = –a if a < 0
|–a| = |a| = a (when a ≥ 0)
Domain: the set of all input values (x) for which the function is defined
Range: the set of all output values (y) the function produces
Function: a relation where every input has exactly one output (passes the vertical line test)
Direct variation: y = kx
Indirect variation: y = k/x
Roots (zeros): the x-values where f(x) = 0
Mean: the sum of all values divided by the number of values
Median: the middle value when data is arranged in order
Mode: the value that appears most frequently
Range: the difference between the greatest and least values (note: this is a different use of "range" from the function definition)
d = √((x₂ – x₁)² + (y₂ – y₁)²)
(m₁, m₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2)
d = rt, where r = rate and t = time
a² + b² = c², where c is the hypotenuse
Given y = mx + b:
A parallel line has the same slope: m
A perpendicular line has slope: –1/m (the negative reciprocal)
To convert between units, multiply by conversion factors so that unwanted units cancel. For example, to convert °C to °F: F = (9/5)C + 32.
The discriminant is b² – 4ac.
If b² – 4ac > 0: two distinct real roots
If b² – 4ac < 0: no real roots (two complex conjugate roots)
If b² – 4ac = 0: exactly one real root (a repeated root)
A piecewise function uses different rules for different intervals of x. Written with a brace grouping the pieces, e.g.:
f(x) = { expression₁ if condition₁; expression₂ if condition₂ }
When graphing, pay attention to open vs closed circles at boundary points.
⚠️ Perfect squares 13² through 25² appear constantly in factoring, simplifying radicals, and the Pythagorean theorem. Memorise them.
⚠️ The quadratic formula is the universal fallback for any quadratic. Do not confuse the sign: it is –b, not +b.
⚠️ Discriminant = 0 means one repeated root, not "no solution." Common mix-up.
⚠️ Parallel lines share the same slope. Perpendicular slopes are negative reciprocals. Students frequently forget the negative.
⚠️ In PEMDAS, multiplication and division have equal precedence (left to right), and so do addition and subtraction. They are not strictly sequential.
Q: What is the slope of a line perpendicular to y = 3x + 7?
A: –1/3 (the negative reciprocal of 3).
Q: What does a discriminant of –4 tell you about the roots of a quadratic?
A: There are no real roots; the equation has two complex conjugate roots.
Q: State the vertex of y = 2(x – 3)² + 5.
A: The vertex is (3, 5).
Q: Evaluate 6! without a calculator.
A: 720.
Q: What is 17²?
A: 289.
Q: Name the three methods for solving a system of equations.
A: Graphing, substitution, and elimination.
algebra 1 review, SYMKC, stuff you must know cold, perfect squares, perfect cubes, powers of 2, factorials, slope-intercept form, point-slope form, standard form, quadratic formula, discriminant, PEMDAS, order of operations, commutative property, associative property, distributive property, absolute value, domain, range, function, direct variation, inverse variation, distance formula, midpoint formula, pythagorean theorem, parallel lines, perpendicular lines, inequality symbols, piecewise functions, systems of equations, central tendency, mean, median, mode