Captain Lugaru

I am a coding Viking Bunny. In 2021 I won internship at ASE Inc.

Ninth Grade Cheat Sheet

Chapter 1: Exponents

Product Rule

Formula: \(X^a \cdot X^b = X^{a+b}\)
Explanation: If you have two numbers with the same base, multiplying them means keeping the base and adding the exponents.
Example: \(5^3 \cdot 5^4 = 5^7\)

Quotient Rule

Formula: \(X^a / X^b = X^{a-b}\), as long as \(x \neq 0\)
Explanation: Divide with same base → subtract exponents.
Example: \(7^5 / 7^2 = 7^3\)

Power of a Power Rule

Formula: \((X^a)^b = X^{a \cdot b}\)
Explanation: Raise a power to another → multiply exponents.
Example: \((3^2)^4 = 3^8\)

Power of a Product Rule

Formula: \((a \cdot b)^x = a^x \cdot b^x\)
Explanation: Distribute the exponent to each factor.
Example: \((2 \cdot 5)^3 = 2^3 \cdot 5^3 = 1000\)

Power of a Quotient Rule

Formula: \((a/b)^x = a^x / b^x\), \(b \neq 0\)
Explanation: Apply exponent to both numerator and denominator.
Example: \((4/3)^2 = 16/9\)

Zero Exponent Rule

Formula: \(X^0 = 1\), \(x \neq 0\)
Explanation: Any nonzero base raised to 0 = 1.
Example: \(10000^0 = 1\)

Negative Exponent Rule

Formula: \(X^{-a} = 1 / X^a\), \(x \neq 0\)
Explanation: Flip base and make exponent positive.
Example: \(5^{-3} = 1/125\); \(1/2^{-4} = 2^4 = 16\)

Exponential Growth and Decay

Growth: \(A = P(1 + r)^t\)
Decay: \(A = P(1 - r)^t\)
Example: \(A = 1000(1.05)^3 = 1157.63\)

Scientific Notation

Used for very large or very small numbers.
Examples:
\(3{,}200{,}000 = 3.2 \times 10^6\)
\(0.00042 = 4.2 \times 10^{-4}\)

Chapter 2: Polynomials

Types of Polynomials

  • Monomial: 1 term (e.g., \(2x\))

  • Binomial: 2 terms (e.g., \(3x + 5\))

  • Trinomial: 3 terms (e.g., \(4x^2 - 3x + 2\))

  • Polynomial: >3 terms (e.g., \(x^4 + 3x^3 - 2x^2 + 5x - 1\))

  • Degree: Highest exponent in the polynomial
    Example: \(4x^3 + 2x^2 - x\) has degree 3

Operations with Polynomials

Collecting Like Terms: Same variables and exponents
Example: \(3x^2 + 5x^2 = 8x^2\)

Distributive Property: \(a(b + c) = ab + ac\)
Example: \(3(x + 4) = 3x + 12\)

Multiplying Polynomials

Binomial × Binomial: \((a + b)(c + d) = ac + ad + bc + bd\)
Example: \((x + 3)(x + 5) = x^2 + 8x + 15\)

Binomial × Trinomial:
\((x + 2)(x^2 + 3x + 4) = x^3 + 5x^2 + 10x + 8\)

Factoring Polynomials

Common Factor:
\(6x^2 + 9x = 3x(2x + 3)\)

Trinomial Factoring:
\(x^2 + 7x + 12 = (x + 3)(x + 4)\)

Chapter 3: Solving Equations

Basic Equations

Example: \(x + 4 = 7\) → \(x = 3\)

Two-Step Equations

Example: \(3x - 5 = 10\) → \(x = 5\)

Equations with Fractions

\(x / 3 = 5\) → Multiply both sides by 3 → \(x = 15\)

Distributive Property in Equations

\(2(x - 3) = 10\) → \(2x - 6 = 10\) → \(x = 8\)

Cross-Multiplication (Proportions)

If \(a/b = c/d\), then \(ad = bc\)
Example: \(x/5 = 3/10\) → \(x = 1.5\)

Variables on Both Sides

Example: \(5x + 3 = 2x + 9\) → \(x = 2\)

Chapter 4: Linear Relationships

Scatter Plots and Correlation

  • Independent Variable → x-axis

  • Dependent Variable → y-axis

  • Correlation Types: Positive, Negative, None

  • Line of Best Fit: Represents data trend

  • Interpolation vs. Extrapolation

Distance-Time Graphs

  • Slope = speed

  • Flat line = no movement

  • Steep = faster

  • Downward = return

  • Curved = acceleration or deceleration

Slope and Equation of a Line

  • Slope: \(m = (y_2 - y_1) / (x_2 - x_1)\)

  • Forms:

    • Slope-Intercept: \(y = mx + b\)

    • Point-Slope: \(y - y_1 = m(x - x_1)\)

    • Standard: \(Ax + By + C = 0\)

  • Direct: \(y = mx\) (through origin)

  • Partial: \(y = mx + b\)

Converting Representations

Table → Equation: Find slope, y-intercept → \(y = mx + b\) Graph → Equation: Use slope and y-intercept → \(y = mx + b\) Equation → Graph: Plot y-intercept, use slope for points

Parallel and Perpendicular Lines

  • Parallel: Same slope

  • Perpendicular: Negative reciprocals

  • Horizontal: \(y = c\) (slope = 0)

  • Vertical: \(x = c\) (slope undefined)

Chapter 5: Geometry

Angles and Triangles

  • Complementary: \(90°\)

  • Supplementary: \(180°\)

  • Parallel Line Pairs: Corresponding, Alternate Interior, Alternate Exterior

Triangle Properties

  • Interior Angles Sum: \(180°\)

  • Exterior Angle = sum of opposite interior angles

  • Isosceles Theorem: Equal sides → equal angles

Right Triangle Properties

Pythagorean Theorem: \(a^2 + b^2 = c^2\) Trigonometry: SOH CAH TOA Angles of Elevation/Depression

Perimeter, Area, and Volume

  • Rectangle: \(P = 2l + 2w\), \(A = lw\)

  • Triangle: \(A = \tfrac{1}{2}bh\)

  • Circle: \(C = 2\pi r\), \(A = \pi r^2\)

  • Sphere: \(V = \tfrac{4}{3}\pi r^3\)

  • Cylinder: \(V = \pi r^2 h\)

Transformations

  • Translation: Slide

  • Reflection: Flip

  • Rotation: Turn

  • Dilation: Resize

Chapter 6: Probability and Statistics

Basic Probability

  • Formula: \(P = \frac{\text{favorable}}{\text{total}}\)

  • Independent: One doesn’t affect the other

  • Dependent: One affects the other

Descriptive Statistics

  • Mean = average

  • Median = middle value

  • Mode = most frequent

  • Range = highest − lowest

Data Representation

  • Bar Graphs

  • Histograms

  • Pie Charts

  • Box Plots

Tenth Grade Cheat Sheet

A reconciliation map between Algebra 1 (the foundation you should already own) and Algebra 2 (where you’re actually headed). Use Algebra 1 as a diagnostic — don’t re-grind what’s already solid.

Each topic has a Practice block. Try them before peeking at any solutions — getting stuck is where the learning happens. If a topic feels trivial, skip the drills and move on.

Algebra 1 — Foundation

Verify with a diagnostic. Anything shaky here will hurt in Algebra 2.

Foundations

Variables, expressions, order of operations, properties of real numbers.

Practice

  1. Evaluate: \(3 + 4 \cdot (2^2 - 5) + 6 \div 2\).

  2. Simplify: \(5(2x - 3) - 2(x - 7)\).

  3. Name the property used: \(a + (b + c) = (a + b) + c\).

  4. Evaluate \(x^3 - 4x + 1\) when \(x = -2\).

  5. Insert parentheses to make the statement true: \(8 - 3 \cdot 2 + 1 = 5\).

Linear Equations & Inequalities

Solving in one variable.

Practice

  1. Solve: \(3(x - 4) = 2x + 5\).

  2. Solve: \(\frac{x}{3} - \frac{x}{4} = 2\).

  3. Solve and graph on a number line: \(-2x + 7 \geq 15\).

  4. Solve: \(|2x - 1| < 5\).

  5. A taxi charges $3 plus $2 per mile. How many miles correspond to a $19 fare? Set up and solve an equation.

Linear Functions & Graphing

Slope, intercepts, and the forms of a line (slope-intercept, point-slope, standard).

Practice

  1. Find the slope of the line through \((-1, 4)\) and \((3, -2)\).

  2. Write the equation (slope-intercept form) of the line through \((2, -1)\) with slope \(3\).

  3. Convert \(2x - 3y = 12\) to slope-intercept form.

  4. Find the line parallel to \(y = \tfrac{1}{2}x + 4\) passing through \((4, 1)\).

  5. A line passes through \((0, -3)\) and \((5, 7)\). Write its equation in standard form.

Systems of Linear Equations

Substitution, elimination, graphing.

Practice

  1. Solve by substitution: \(y = 2x + 1\) and \(3x + y = 11\).

  2. Solve by elimination: \(2x + 3y = 12\) and \(4x - y = 10\).

  3. Two numbers sum to 30 and differ by 8. Find them.

  4. Solve: \(x + y + z = 6\), \(2x - y + z = 3\), \(x + 2y - z = 5\).

  5. How many solutions does the system \(4x - 2y = 6\), \(2x - y = 3\) have? Justify.

Exponents

Exponent rules and manipulation.

Practice

  1. Simplify: \((3x^2 y^3)(2x^4 y)\).

  2. Simplify: \(\dfrac{a^5 b^2}{a^2 b^5}\).

  3. Evaluate: \(2^{-3} + 3^0\).

  4. Simplify: \((2x^3)^4\).

  5. Solve for \(n\): \(2^n = 32\).

Polynomials

Adding, subtracting, multiplying.

Practice

  1. Add: \((3x^2 - 2x + 1) + (x^2 + 4x - 6)\).

  2. Multiply: \((2x - 3)(x + 5)\).

  3. Expand: \((x - 4)^2\).

  4. Subtract: \((5x^3 - x + 2) - (2x^3 + 3x^2 - 1)\).

  5. Multiply: \((x + 2)(x^2 - 3x + 4)\).

Factoring

The big one. GCF, trinomials, difference of squares.

Practice

  1. Factor: \(x^2 + 7x + 12\).

  2. Factor: \(x^2 - 49\).

  3. Factor: \(6x^2 + 11x - 10\).

  4. Factor by grouping: \(x^3 - 2x^2 + 3x - 6\).

  5. Factor completely: \(3x^3 - 27x\).

Quadratic Equations & Functions

Factoring, the quadratic formula, parabolas.

Practice

  1. Solve by factoring: \(x^2 - 5x + 6 = 0\).

  2. Solve with the quadratic formula: \(2x^2 + 3x - 5 = 0\).

  3. Find the vertex of \(y = x^2 - 6x + 11\).

  4. A ball follows \(h(t) = -16t^2 + 32t + 6\). When does it hit the ground?

  5. Use the discriminant to determine how many real roots \(3x^2 - 4x + 2 = 0\) has.

Radicals & Rational Expressions

Basic operations and simplification.

Practice

  1. Simplify: \(\sqrt{72}\).

  2. Rationalize: \(\dfrac{3}{\sqrt{5}}\).

  3. Simplify: \(\dfrac{x^2 - 9}{x^2 - x - 6}\).

  4. Add: \(\dfrac{1}{x} + \dfrac{2}{x + 1}\).

  5. Solve: \(\sqrt{2x + 3} = 5\).

Intro to Data & Statistics

Sometimes tacked on at the end.

Practice

  1. Find the mean, median, and mode of: \(4, 7, 7, 9, 12, 15\).

  2. Compute the range and the interquartile range of the data above.

  3. A class has mean \(80\), standard deviation \(5\). A student scored \(90\). How many standard deviations above the mean is she?

  4. A bag has 3 red and 5 blue marbles. What is the probability of drawing red?

  5. Plot \((1,2), (2,4), (3,5), (4,8)\) on a scatter plot. Describe the correlation and estimate a line of best fit.

Algebra 2 — Where You’re Headed

Equations & Inequalities Review

Fast pass — Algebra 1 carryover.

Practice

  1. Solve: \(5 - 2(3x - 4) = 7\).

  2. Solve: \(|3x - 2| = 7\).

  3. Solve: \(2x^2 - 5x - 3 = 0\).

  4. Solve the inequality: \(x^2 - x - 6 < 0\).

  5. Solve the system: \(2x + y = 5\) and \(x^2 + y = 7\).

Functions

Notation, domain/range, transformations, inverses, composition.

Practice

  1. If \(f(x) = 2x - 3\) and \(g(x) = x^2\), find \((f \circ g)(2)\).

  2. Find the inverse of \(f(x) = \dfrac{x - 4}{3}\).

  3. State the domain of \(f(x) = \dfrac{\sqrt{x - 2}}{x - 5}\).

  4. Describe the transformations taking \(f(x) = x^2\) to \(g(x) = -2(x + 3)^2 + 1\).

  5. If \(f(x) = \dfrac{1}{x - 2}\), find \(f(f(3))\).

Quadratics & Complex Numbers

Imaginary numbers enter here.

Practice

  1. Simplify: \((3 + 2i)(4 - i)\).

  2. Solve: \(x^2 + 4x + 13 = 0\).

  3. Express in standard form: \(\dfrac{2 + i}{1 - i}\).

  4. Find the modulus \(|3 - 4i|\).

  5. Solve: \(x^2 = -16\).

Polynomial Functions

Higher-degree polynomials, polynomial division, roots, factor and remainder theorems.

Practice

  1. Use long division: \((2x^3 - 3x^2 + x - 5) \div (x - 2)\).

  2. Use the Remainder Theorem to find \(P(-1)\) for \(P(x) = x^4 - 3x^2 + 2\).

  3. Find all rational roots of \(x^3 - 2x^2 - 5x + 6\).

  4. Describe the end behavior of \(f(x) = -2x^5 + 3x^2 - 1\).

  5. Write a polynomial of degree 4 (real coefficients) with roots \(1\), \(-2\), and \(3i\).

Rational Expressions & Functions

Asymptotes show up.

Practice

  1. Simplify: \(\dfrac{x^2 - 4}{x^2 + 4x + 4}\).

  2. Find all asymptotes of \(f(x) = \dfrac{2x^2 - 3}{x^2 - 1}\).

  3. Solve: \(\dfrac{1}{x - 1} + \dfrac{2}{x} = 3\).

  4. Multiply and simplify: \(\dfrac{x^2 - 9}{x + 2} \cdot \dfrac{x + 2}{x - 3}\).

  5. Identify any holes vs vertical asymptotes in \(f(x) = \dfrac{x^2 - x - 6}{x - 3}\).

Radical Functions & Rational Exponents

Connecting roots to fractional exponents.

Practice

  1. Simplify: \(x^{1/2} \cdot x^{2/3}\).

  2. Solve: \(\sqrt{x + 7} = x - 5\) (check for extraneous solutions).

  3. Rewrite \(x^{3/4}\) in radical form.

  4. State the domain of \(f(x) = \sqrt{2x - 6}\).

  5. Simplify: \(\sqrt[3\){54 x^4}].

Exponential Functions

Growth, decay, base-e.

Practice

  1. Solve: \(3^{x+1} = 81\).

  2. A population doubles every 5 years. If it is 1,000 now, what is it in 20 years?

  3. Sketch \(f(x) = 2^x\) and \(g(x) = (1/2)^x\) on the same axes; compare.

  4. Compute the value of $1,000 invested at 5% continuous interest for 10 years using \(A = Pe^{rt}\).

  5. Solve: \(e^{2x} = 7\).

Logarithms

The classic wall. Budget extra time.

Practice

  1. Evaluate: \(\log_2 32\).

  2. Solve: \(\log(x) + \log(x - 3) = 1\).

  3. Use change of base to evaluate \(\log_5 20\).

  4. Expand: \(\log\!\left(\dfrac{x^2 y^3}{z}\right)\).

  5. Solve: \(5^x = 12\).

Sequences & Series

Arithmetic, geometric, summation notation.

Practice

  1. Find the 20th term of the sequence \(3, 7, 11, 15, \ldots\).

  2. Find the sum of the first 15 terms of \(2, 6, 18, 54, \ldots\).

  3. Write in summation notation: \(1 + 4 + 9 + 16 + 25\).

  4. Find the sum of the infinite series \(1 + \tfrac{1}{3} + \tfrac{1}{9} + \cdots\).

  5. Prove that \(0.\overline{9} = 1\) using a geometric series.

Conic Sections

Circles, ellipses, parabolas, hyperbolas.

Practice

  1. Write the equation of the circle with center \((-2, 3)\) and radius \(5\).

  2. Identify the conic and its key features: \(x^2 + 4y^2 = 16\).

  3. Find the foci of \(\dfrac{x^2}{25} - \dfrac{y^2}{9} = 1\).

  4. Convert to standard form: \(x^2 + y^2 - 6x + 4y - 12 = 0\).

  5. Find the focus and directrix of \(y = \tfrac{1}{8}x^2\).

Probability & Statistics

Often the closing unit.

Practice

  1. A fair coin is flipped 4 times. What is the probability of exactly 3 heads?

  2. In how many ways can 5 distinct books be arranged on a shelf?

  3. Two dice are rolled. What is the probability the sum is 7?

  4. From a class of 20 students, how many ways can a committee of 3 be chosen?

  5. Test scores are normally distributed with \(\mu = 70\), \(\sigma = 10\). What is the probability a score exceeds 85? (Use a z-table.)

Trigonometry Intro

Some Algebra 2 courses include the basics; others push it to Pre-Calc.

Practice

  1. In a right triangle with legs 3 and 4, find \(\sin\), \(\cos\), and \(\tan\) of the smaller acute angle.

  2. Convert \(150°\) to radians.

  3. Evaluate: \(\sin(\pi/6)\), \(\cos(\pi/4)\), \(\tan(\pi/3)\).

  4. In a 30-60-90 triangle with hypotenuse 10, find the two legs.

  5. In a right triangle, one acute angle is \(35°\) and the opposite side is 12. Find the adjacent side (use a calculator).

Resources — How To Actually Learn This

Picked for depth over polish. Khan Academy is here for gap-checking only — it’s procedural and shallow on the why. The rest treat you like a mathematician.

Video Courses

Resource Cost Why this one

Professor Leonard — Intermediate & College Algebra (YouTube)

Free

Full lecture series. Derives everything, no hand-waving. Best single video resource for Algebra 2 end-to-end.

3Blue1Brown — Lockdown Math, Essence of Linear Algebra, complex number videos

Free

Geometric intuition behind symbol manipulation. Watch his logarithm and complex number videos when those walls hit.

MIT OCW — Highlights of Calculus (Gilbert Strang)

Free

Strang explains structure, not steps. Shows algebra in service of higher math.

Khan Academy — Algebra 1 & 2

Free

Use as a diagnostic and fluency drill. Not a primary text — it’s procedural.

Textbooks

Resource Cost Why this one

Gelfand — Algebra

~$25

Slim, rigorous, every problem earns its place. Written by a Fields-tier mathematician for serious students. Start here.

Gelfand — Functions and Graphs + The Method of Coordinates

~$20 each

Companions to Algebra. Build the geometric half — functional thinking, conics, coordinate geometry — properly.

Lang — Basic Mathematics

~$50

Algebra 1 through pre-calc as one unified rigorous whole. Lang writes algebra the way he’d write it for future mathematicians.

AoPS — Introduction to Algebra + Intermediate Algebra (Rusczyk)

~$60 each

The gold standard for above-grade-level students. Full Algebra 1+2 coverage with problems that force genuine thinking.

Problem-Solving & Practice

Resource Cost Why this one

AoPS Alcumus

Free (account)

Adaptive problem engine. Difficulty scales with you — not a worksheet generator.

Brilliant — Algebra courses

~$120/yr

Interactive, problem-first. Good on complex numbers, functions, sequences.

Paul’s Online Math Notes (Lamar University)

Free

Terse, accurate reference. Best for clean lookups on logs, conics, partial fractions.

Algebra 2 Walls — Targeted Help

Topic Where to go

Logarithms

Gelfand Functions and Graphs (exponentials chapter) + 3Blue1Brown "Logarithm fundamentals."

Conic sections

Gelfand The Method of Coordinates + AoPS Intermediate Algebra ch. 14.

Complex numbers

Needham Visual Complex Analysis ch. 1 (older edition PDF free online) + 3Blue1Brown "Complex number fundamentals." Needham is geometric, rigorous, and made for someone who reads Gödel for fun.

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