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G = 6.67 × 10⁻¹¹ N·m²/kg² (gravitational constant)
g = 9.8 m/s² or 9.8 N/kg (gravity at Earth's surface)
1 atm = 1.0 × 10⁵ Pa
Prefixes: T (10¹²), G (10⁹), M (10⁶), k (10³), c (10⁻²), m (10⁻³), µ (10⁻⁶), n (10⁻⁹), p (10⁻¹²)
Trig Values: sin 0°=0, sin 30°=1/2, sin 37°=3/5, sin 45°=√2/2, sin 53°=4/5, sin 60°=√3/2, sin 90°=1
Geometry: A=πr² (circle), C=2πr, V=4/3πr³ (sphere), a²+b²=c² (right triangle)
v = v₀ + at
Δx = v₀t + ½at²
v² = v₀² + 2aΔx
x_cm = Σmᵢxᵢ / Σmᵢ (center of mass)
F_net = ma = m(Δv/Δt) = Δp/Δt
F_g = Gm₁m₂/r²
F_f ≤ µF_n (friction)
F_s = -kΔx (Hooke's law — spring)
a_c = v²/r (centripetal acceleration)
K = ½mv² (kinetic energy)
W = Fd cos θ (work)
ΔK = ΣW (work-energy theorem)
ΔU_s = ½k(Δx)² (spring PE)
ΔU_g = mgΔy (gravitational PE near surface)
P_avg = W/Δt = ΔE/Δt (average power)
p = mv (linear momentum)
J = F_avg Δt = Δp (impulse-momentum theorem)
ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt² (rotational kinematics)
τ = rF sin θ (torque)
I = Σmr² (rotational inertia)
K = ½Iω² (rotational KE)
L = Iω or L = rmv sin θ (angular momentum)
ΔL = τΔt (angular impulse)
T_s = 2π√(m/k) (spring period)
T_p = 2π√(L/g) (pendulum period)
ρ = m/V (density)
P = F/A, P = P₀ + ρgh (pressure)
F_b = ρVg (buoyant force)
A₁v₁ = A₂v₂ (continuity equation)
P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂² (Bernoulli's equation)
Ch 2: One-Dimensional Kinematics
Kinematics is the study of motion without considering its causes. Displacement Δx = xf − x0 is the change in position (vector). Distance is the magnitude of displacement. The SI unit is meters (m).
Vectors have magnitude + direction; scalars have magnitude only (e.g., speed is scalar, velocity is vector).
Average velocity v̄ = Δx/Δt. Average speed = total distance / total time (not the same as magnitude of avg velocity!).
Acceleration a = Δv/Δt — rate of change of velocity. SI unit: m/s². An object in free fall accelerates at g = 9.8 m/s² downward.
Key Equations (constant acceleration):
Problem-solving tip: Assign a positive direction first. g is always 9.8 m/s² downward. For objects thrown upward, g is negative relative to positive up.
Ch 3: Two-Dimensional Kinematics
Motion in 2D can be analyzed by separating into independent horizontal and vertical components, each analyzed separately with the 1D kinematic equations.
Vector addition: Use head-to-tail method (graphical) or component method (analytical). Components: A_x = A cos θ, A_y = A sin θ. Magnitude: R = √(R_x² + R_y²).
Projectile motion: Horizontal motion: constant velocity (a_x = 0). Vertical motion: constant acceleration (a_y = −g = −9.8 m/s²). The horizontal and vertical components are independent. Range is maximized at 45° launch angle.
Add the OpenStax link to your resources above for the full text!
Ch 4: Dynamics — Newton's Laws of Motion
Force is a push or pull (vector). Dynamics is the study of how forces affect motion.
Newton's 1st Law (Inertia): A body at rest stays at rest; a body in motion stays in motion at constant velocity — unless acted on by a net external force. Inertia is the tendency of an object to resist change in motion; it's related to mass.
Newton's 2nd Law: a = F_net / m, or more familiarly F_net = ma. The acceleration is directly proportional to net force, inversely proportional to mass, and in the same direction as the net force.
Weight w = mg — the force of gravity on an object.
Newton's 3rd Law: For every action force, there is an equal and opposite reaction force. If object A exerts force on object B, B exerts a force equal in magnitude and opposite in direction back on A.
Common forces: Normal force (perpendicular to surface, supports weight), Tension (force along a rope/string/cable), Friction (opposes motion, parallel to surface).
Free-body diagram: Draw the object, then draw ALL external forces acting on it as vectors. This is the #1 most important problem-solving skill.
Ch 5: Friction, Drag & Elasticity
Friction: Static friction f_s ≤ μ_s N (surfaces not sliding). Kinetic friction f_k = μ_k N (surfaces sliding). μ_s > μ_k usually. Friction always opposes motion.
Drag force F_D = ½CρAv² (for larger objects in fluid). For small objects in dense fluids: Stokes' law F_s = 6πηrv.
Hooke's law (elasticity): F = kΔL. Young's modulus: stress = Y × strain. Shear modulus and bulk modulus for other deformation types.
Uniform Circular Motion: Motion in a circle at constant speed. The direction of velocity is always tangent to the circle — it's constantly changing, so the object is accelerating even though speed is constant!
Rotation angle: Δθ = Δs/r (in radians). 2π rad = 360°.
Angular velocity: ω = Δθ/Δt (rad/s). Linear speed: v = rω.
Centripetal acceleration: a_c = v²/r = rω² — always points toward the center of the circle ("center-seeking").
Centripetal force: F_c = ma_c = mv²/r = mrω² — the net force toward the center that causes circular motion. NOT a separate force — it's provided by tension, gravity, friction, normal force, etc.
Newton's Universal Law of Gravitation: Every particle attracts every other particle with F = Gm₁m₂/r². G = 6.67×10⁻¹¹ N·m²/kg².
Gravitational field strength: g = F/m = GM/r². At Earth's surface: g = 9.8 N/kg.
Kepler's Laws (for satellites/planets): (1) Elliptical orbits with Sun at one focus. (2) Equal areas in equal times. (3) T² ∝ r³ (period squared ∝ orbital radius cubed).
Orbital velocity: For a circular orbit: v = √(GM/r). Centripetal force is provided by gravity: GMm/r² = mv²/r.
Work: W = Fd cos θ. Work = force × displacement × cos of the angle between them. Only the force component parallel to displacement does work. SI unit: joule (J) — 1 J = 1 N·m. Positive work increases energy; negative work decreases energy.
Kinetic Energy: K = ½mv² — energy of motion.
Work-Energy Theorem: The net work done on a system equals its change in kinetic energy: W_net = ΔK = ½mv_f² − ½mv_i².
Potential Energy: Stored energy due to position.
Conservative Forces: Work depends only on start/end points, not path (gravity, spring force). For any conservative force, we can define potential energy.
Conservation of Mechanical Energy: When ONLY conservative forces act: KE + PE = constant. KE_i + PE_i = KE_f + PE_f.
Non-conservative forces (friction, drag): dissipate energy as heat. If they act: W_nc = ΔKE + ΔPE.
Power: Rate of doing work. P_avg = W/Δt = ΔE/Δt. P = Fv cos θ (instantaneous). SI unit: watt (W) — 1 W = 1 J/s.
Linear Momentum: p = mv. SI unit: kg·m/s. Momentum is a vector in the same direction as velocity.
Newton's 2nd Law (momentum form): F_net = Δp/Δt. Net force = rate of change of momentum.
Impulse: J = F_avg Δt = Δp = mΔv. The product of force and time equals the change in momentum. To reduce force in a collision, increase the time (airbags, crumple zones).
Conservation of Momentum: In an ISOLATED system (F_net = 0), total momentum is conserved: p_total_initial = p_total_final. This is one of the most powerful principles in physics!
For projectile motion (no air resistance): Momentum is conserved in the horizontal direction only (no horizontal forces).
Collisions:
For 2D collisions, break momentum into x and y components. Conservation applies to each component separately.
Simple Harmonic Motion (SHM): Oscillating motion where the restoring force is proportional to the displacement from equilibrium (F = −kx — Hooke's law).
Period & Frequency: T = 1/f. T = time for one complete cycle (seconds). f = cycles per second (Hz).
Mass-Spring System: T_s = 2π√(m/k). The period depends on mass and spring constant. A stiffer spring (larger k) gives shorter period; larger mass gives longer period.
Simple Pendulum: T_p = 2π√(L/g). For small angles only! The period does NOT depend on mass or amplitude — only on length and gravity. Longer string = longer period.
Position vs. Time: x(t) = A cos(2πft) or x(t) = A sin(2πft). A = amplitude (maximum displacement).
Energy in SHM: Total mechanical energy is constant. At equilibrium: max KE, min PE. At amplitude: min KE, max PE (spring PE = ½kA²).
Rotational Kinematics (Ch 10): Analogous to linear kinematics:
Relationships: v = rω (linear speed from angular speed). a_T = rα (tangential acceleration). Δx_cm = rΔθ (arc displacement for rolling).
Torque (Ch 9): τ = r_⊥F = rF sin θ — a force's tendency to cause rotation. The lever arm (r_⊥) is the perpendicular distance from the pivot to the line of action of the force. Direction: clockwise or counterclockwise.
Rotational Inertia: I = Σmᵢrᵢ² — the rotational equivalent of mass. Depends on mass distribution relative to the axis. Parallel axis theorem: I = I_cm + Md².
Newton's 2nd Law (Rotation): α = τ_net / I — net torque = rotational inertia × angular acceleration.
Rotational KE: K_rot = ½Iω².
Angular Momentum: L = Iω (rigid body) or L = rmv sin θ (point particle). Conservation: if no net external torque, angular momentum is conserved. τ = ΔL/Δt (angular impulse-momentum theorem).
Statics (Ch 9): For an object in equilibrium: ΣF = 0 (translational equilibrium) AND Στ = 0 (rotational equilibrium).
Density: ρ = m/V (kg/m³).
Pressure: P = F/A (Pa = N/m²). P_absolute = P₀ + ρgh (at depth h). P_gauge = ρgh (difference from atmospheric).
Buoyancy (Archimedes' Principle): F_b = ρVg — the buoyant force equals the weight of the fluid displaced. If F_b > weight → object floats. If F_b < weight → object sinks. An object floats when its density < fluid density.
Fluid Flow (Ch 12):
Rhetorical Situation: Every text exists within a specific moment — you must analyze who is speaking, to whom, why now, and under what circumstances.
Speaker (or Writer): Not the same as the author — consider the persona/voice being constructed. What ethos does the speaker establish? Are they credible, passionate, objective?
Purpose: What does the writer want to achieve? To persuade, inform, entertain, inspire, provoke? Every rhetorical choice serves the purpose.
Audience: Who is the intended reader/listener? Consider demographics, values, expectations, and prior knowledge. A text aimed at experts looks very different from one for the general public.
Context & Exigence: Context = the historical/cultural moment surrounding the text. Exigence = the specific issue or problem that prompts the writing. The exigence is the "why now?" — the urgent need the text responds to.
AP Tip: The rhetorical situation is the foundation of every analysis. Start by identifying speaker, purpose, audience, context, and exigence before diving into specific devices.
Thesis Statement: A defensible claim that responds to the prompt. A strong thesis does NOT just restate the prompt — it takes a position that requires support and reasoning to prove.
Types of Claims:
Using Evidence: Evidence must be relevant, sufficient, and accurately represented. Types: facts/stats, examples/anecdotes, expert testimony, analogy, textual evidence (quotations). Don't just drop evidence — comment on it and explain HOW it supports your claim.
Synthesis (FRQ 1): You must use at least 3 of the provided sources. Don't just summarize — engage with them. Use sources as support, counterargument, or complication. Cite properly (author/title is fine for AP).
AP Tip: For the argument essay (FRQ 3), the best evidence comes from your own knowledge. Use specific, well-developed examples from history, literature, current events, or personal experience.
Line of Reasoning: The logical progression from claim to conclusion. Each paragraph should connect to the next, building a cumulative case. You should be able to summarize your argument in a few sentences — that's your line of reasoning.
Methods of Development (how to organize):
Transitions: Don't just list your points — connect them. Use transitions to show relationships: however (contrast), therefore (conclusion), similarly (comparison), moreover (addition), for example (illustration).
AP Tip: The rhetorical analysis essay (FRQ 2) scores heavily on your ability to trace the writer's line of reasoning — how each choice builds toward the purpose.
Diction (Word Choice): Formal vs. informal? Abstract vs. concrete? Denotation (dictionary meaning) vs. connotation (emotional associations). Why THIS word instead of a synonym? Every word choice is deliberate.
Syntax (Sentence Structure): Short sentences = urgency, emphasis. Long sentences = complexity, reflection. Sentence variety keeps readers engaged. Key patterns to look for:
Tone: The writer's attitude toward the subject/audience. Described with adjectives: sarcastic, reverent, urgent, detached, earnest, ironic. Tone is created through a combination of diction, syntax, and imagery.
Figurative Language: Metaphor (direct comparison), simile (like/as), personification, hyperbole, allusion (reference to history/literature/bible), irony (verbal, situational, dramatic). Each device serves the purpose — don't just identify it, explain WHY.
AP Tip: In the rhetorical analysis essay, go beyond listing devices. Explain the EFFECT of each device on the audience and how it advances the writer's purpose.
🌎 Native American Diversity: Before Europeans, the Americas were filled with diverse societies. In the Southwest, the Pueblo peoples built irrigation systems and lived in adobe villages. The Great Basin & Plains tribes (like the Sioux) were largely nomadic, following buffalo herds. In the Northeast, the Iroquois League formed a powerful confederation of 5 (later 6) tribes. Along the Mississippi, the Cahokia built massive mound cities. Key idea: no single "Native American" culture — each was shaped by their environment.
🇪🇸 European Motivations: Spain led the way with Columbus (1492). Three Gs: Gold (wealth), God (spread Catholicism), Glory (empire). The Spanish established the encomienda system — forced Native labor on plantations/mines in exchange for "protection" and Christian instruction. Bartolomé de las Casas criticized this brutality.
🤝 Columbian Exchange: Massive transfer of goods, people, diseases, and ideas between Old World and New World. From Americas to Europe: corn, potatoes, tomatoes, tobacco, chocolate. From Europe to Americas: horses, cattle, wheat, sugar, and DEVASTATING diseases (smallpox, measles) that killed 90% of Native populations — the single greatest demographic catastrophe in history.
⚔️ Spanish Empire: Conquistadors like Cortés (Aztecs) and Pizarro (Inca) conquered massive empires with superior weapons, disease, and Native allies who resented Aztec/Inca rule. Spain created a racial caste system: peninsulares (born in Spain) → criollos (Spanish born in Americas) → mestizos (mixed) → Natives → enslaved Africans.
🔗 Links:
🎥 Heimler's Period 1 Review🏴 Chesapeake Colonies (VA & MD): Founded for economic reasons. Jamestown (1607) — first permanent English settlement. Initially a disaster — "starving time" — saved by John Smith's leadership and later by tobacco (John Rolfe). Tobacco created huge demand for land and labor. Led to the headright system (50 acres per person) and indentured servitude. Eventually, Bacon's Rebellion (1676) — poor frontiersmen vs. wealthy elite — led planters to shift from indentured servants to enslaved Africans.
⛪ New England Colonies (MA, CT, RI, NH): Founded for religious reasons. Pilgrims at Plymouth (1620) — Mayflower Compact (early self-gov). Puritans under John Winthrop established Massachusetts Bay Colony — "city upon a hill" (model Christian society). Puritan society: focused on literacy (to read the Bible), church membership required for voting. Half-Way Covenant (1662) — allowed partial membership for children of members who hadn't had conversion experience. Salem Witch Trials (1692) — mass hysteria reflecting social tensions.
🌾 Middle Colonies (NY, NJ, PA, DE): "Breadbasket" colonies — wheat, diverse farming. Pennsylvania founded by William Penn (Quaker) — religious tolerance, fair treatment of Natives. Most ethnically/religiously diverse region.
🛳️ Southern Colonies (NC, SC, GA): Plantation economy — rice, indigo. Relied heavily on enslaved labor. Georgia founded as a debtor colony and buffer against Spanish Florida.
💡 Salutary Neglect: Britain loosely enforced trade laws (Navigation Acts) in exchange for colonial economic loyalty. This allowed colonies to develop their own self-governing traditions (colonial assemblies, town meetings).
📜 Key Colonial Conflicts: Bacon's Rebellion (1676), King Philip's War (1675-78) — Native resistance in New England. The First Great Awakening (1730s-40s) — religious revival, leaders like Jonathan Edwards ("Sinners in the Hands of an Angry God") and George Whitefield; challenged established church authority.
🎥 Heimler's Period 2 Review⚔️ French & Indian War (1754-1763): Britain vs. France (both with Native allies) over Ohio River Valley. Britain won — got French Canada and Florida in the Treaty of Paris (1763). BUT Britain was deep in debt. This led them to tax the colonies directly, ending salutary neglect. Colonists were furious — they had no representation in Parliament ("no taxation without representation"). The Proclamation of 1763 banned settlement west of Appalachians — colonists felt cheated.
💥 Road to Revolution: Sugar Act (1764), Stamp Act (1765) — first direct tax — led to Stamp Act Congress. Declaratory Act (1766) — Parliament CAN tax colonies. Townshend Acts (1767) — taxes on glass, lead, paper, tea. Boston Massacre (1770). Tea Act (1773) — Boston Tea Party. Intolerable Acts (1774) — closed Boston port, revoked MA charter. First Continental Congress — boycotted British goods.
🔫 American Revolution (1775-1783): Lexington & Concord ("shot heard round the world"). Second Continental Congress — created Continental Army, appointed Washington as commander. Common Sense (Paine, 1776) — argued for independence. Declaration of Independence (July 4, 1776) — Jefferson's Enlightenment-inspired document: "all men are created equal" with natural rights (life, liberty, pursuit of happiness). Key battles: Saratoga (turning point — French alliance), Yorktown (final victory).
📜 Articles of Confederation (1781-1788): FIRST government — intentionally WEAK (feared strong central power). No executive, no courts, no power to tax. Couldn't regulate interstate commerce or raise army. Shays' Rebellion (1786-87) — farmers rebelled against debt/foreclosure — showed Articles were too weak. Led to Constitutional Convention (1787).
📋 Constitution (1787/1788): Great Compromise — bicameral Congress (House by population, Senate equal). 3/5 Compromise — enslaved counted as 3/5 for representation. Separation of powers (3 branches) with checks and balances. Federalism (national + state power). Federalists (Hamilton, Madison, Jay — Federalist Papers) vs. Anti-Federalists (fear of centralized power, wanted Bill of Rights). Bill of Rights (1791) — first 10 amendments protecting individual liberties.
👑 Early Republic Challenges: Washington set precedents (2-term limit, cabinet). Hamilton's Financial Plan — national bank, assumption of state debts, tariffs. Whiskey Rebellion (1794) — showed federal power to suppress revolt. Washington's Farewell Address — warned against political parties and foreign entanglements. Adams administration — XYZ Affair, Alien & Sedition Acts, Kentucky/Virginia Resolutions (states' rights/compact theory).
🎥 Heimler's Period 3 Review🗳️ Jeffersonian Era (1800-1824): "Revolution of 1800" — first peaceful transfer of power between rival parties. Jefferson bought Louisiana Territory (1803) — doubled US size. Lewis & Clark expedition. Marbury v. Madison (1803) — established judicial review (Supreme Court can strike down laws). Embargo Act (1807) — disaster, hurt US economy more than Britain/France. War of 1812 — "Second War of Independence" — ended impressment, boosted American nationalism.
🏛️ Era of Good Feelings (1815-1825): Monroe presidency — one-party rule (Democratic-Republicans). American System (Henry Clay) — national bank, protective tariffs, internal improvements (roads/canals). Missouri Compromise (1820) — MO = slave, ME = free, 36°30' line for future states. Monroe Doctrine (1823) — warned Europe not to colonize/interfere in Western Hemisphere.
👢 Jacksonian Democracy (1828-1840): Andrew Jackson — "common man" president. Spoils system (reward political supporters). Indian Removal Act (1830) — forced relocation of Natives (Trail of Tears — Cherokee). Nullification Crisis — SC said it could nullify federal tariffs; Jackson threatened force. Vetoed the Second Bank of the US — "Bank War."
🏭 Market Revolution: Rapid economic transformation — factories, canals (Erie Canal), railroads, steamboats. Northern industrialization vs. Southern cotton boom (Eli Whitney's cotton gin made slavery MORE profitable, not less). Rise of textile mills in Lowell, MA — "Lowell Girls."
✨ Second Great Awakening & Reform: Religious revival — emphasized individual salvation, social reform. Sparked: Abolitionism (Garrison, Douglass, Grimké sisters), Women's Rights (Seneca Falls Convention 1848 — Declaration of Sentiments), Temperance, Education reform (Horace Mann), Transcendentalism (Emerson, Thoreau). Utopian communities (Oneida, Brook Farm, Shakers).
🎥 Heimler's Period 4 Review🤠 Manifest Destiny (1844-1850s): Belief that America was destined to expand coast to coast. Annexation of Texas (1845). Mexican-American War (1846-1848) — US won and got CA, AZ, NM, UT, NV, CO, WY (Treaty of Guadalupe Hidalgo). Wilmot Proviso (proposed ban on slavery in new territories) — failed, but exposed deep sectional divide. Oregon Trail, California Gold Rush (1849).
⚡ Crisis & Compromise (1850s): Compromise of 1850 — CA = free, popular sovereignty for rest, stricter Fugitive Slave Act. Kansas-Nebraska Act (1854) — repealed Missouri Compromise, popular sovereignty for KS/NE — led to "Bleeding Kansas" (pro-slavery vs. anti-slavery violence). Dred Scott v. Sandford (1857) — Supreme Court said enslaved = property, Congress couldn't ban slavery in territories. Lincoln-Douglas Debates (1858) — Lincoln argued slavery was wrong; Douglas promoted popular sovereignty.
💥 Secession & Civil War (1861-1865): John Brown's raid on Harpers Ferry (1859). Lincoln elected (1860) — Southern states seceded, formed Confederate States of America. Fort Sumter (April 1861) — war begins. North advantages: more people, industry, railroads, navy. South advantages: better generals, fighting on home ground, motivated by defense of way of life. Emancipation Proclamation (Jan 1, 1863) — freed enslaved in rebel states, shifted war goal to ending slavery. Key battles: Antietam, Gettysburg (turning point), Vicksburg. Sherman's March to the Sea. Surrender at Appomattox (April 1865). Lincoln assassinated.
🔄 Reconstruction (1865-1877): Lincoln's 10% Plan vs. Radical Republicans' Wade-Davis Bill. 13th Amendment (abolished slavery). Freedmen's Bureau — helped formerly enslaved with education/jobs. Black Codes — Southern laws restricting Black freedom. 14th Amendment (birthright citizenship, equal protection). 15th Amendment (voting for Black men). Radical Reconstruction — military districts in South. Sharecropping — economic system that kept Black farmers in debt/poverty. Compromise of 1877 — ended Reconstruction, removed federal troops from South. Led to Jim Crow segregation and Plessy v. Ferguson ("separate but equal").
🎥 Heimler's Period 5 Review🏗️ Industrialization: Explosive industrial growth — railroads (Transcontinental RR 1869), steel (Bessemer process), oil (Rockefeller — Standard Oil), electricity (Edison, Westinghouse). Robber Barons or Captains of Industry? Rockefeller — horizontal integration (buying competitors). Carnegie — vertical integration (controlling all steps of production). JP Morgan — banking/finance. Laissez-faire — government stayed out of business. Social Darwinism — justified wealth inequality as "survival of the fittest."
👷 Labor Movements: Horrible working conditions — long hours, low pay, child labor. Knights of Labor (all workers, including women and Black). AFL (Samuel Gompers) — skilled workers, focused on better wages/hours. Major strikes: Great Railroad Strike (1877), Haymarket Affair (1886) (bomb at labor rally — set back labor movement), Homestead Strike (1892), Pullman Strike (1894) (Eugene V. Debs — later socialist leader).
🌵 Westward Expansion & Conflict: Homestead Act (1862) — free land for settlers. Transcontinental RR accelerated settlement. Dawes Act (1887) — broke up tribal lands, forced assimilation (ended up losing more Native land). Wounded Knee Massacre (1890) — end of Indian Wars. Cattle kingdom, farming on Great Plains — hard life, but mythologized.
🌾 Populist Movement: Farmers struggling — debt, railroad monopoly pricing. Granger Movement → Farmers' Alliance → People's (Populist) Party (1890s). Demanded: free silver (inflation to help debtors), income tax, direct election of senators, 8-hour workday. Election of 1896 — William Jennings Bryan ("Cross of Gold" speech) vs. McKinley. McKinley won — populism faded, but many ideas later adopted.
🌆 Urbanization & Immigration: Cities exploded — New York, Chicago. "New Immigration" — Southern and Eastern Europeans (Italians, Poles, Jews, Russians). Lived in ethnic enclaves, faced nativist discrimination. Chinese Exclusion Act (1882) — first federal law restricting immigration by ethnicity. Urban problems: tenements, sanitation, political machines (Tammany Hall — Boss Tweed). Settlement houses (Jane Addams — Hull House).
🎥 Heimler's Period 6 Review⭐ Progressive Era (1890-1920): Reform movement to fix problems of Gilded Age — corruption, monopoly, social injustice. Muckrakers — investigative journalists (Ida Tarbell — Standard Oil, Upton Sinclair — The Jungle → Pure Food & Drug Act). TR's Square Deal — trust-busting (Northern Securities), consumer protection, conservation (national parks). 17th Amendment (direct election of senators). 19th Amendment (women's suffrage). Wilson's New Freedom — Federal Reserve (1913), Clayton Antitrust Act. Limits: Prohibition (18th Amendment), eugenics, racism within movement.
🌍 American Imperialism: US became a world power. Spanish-American War (1898) — got Philippines, Guam, Puerto Rico, Cuba (Platt Amendment). Open Door Policy (China). Panama Canal (1914). Roosevelt Corollary — US could intervene in Latin America. Anti-imperialists opposed (too expensive, violated American principles).
💣 WWI (1914-1918): US neutral initially, but: unrestricted submarine warfare (Lusitania), Zimmermann Telegram. US entered 1917 — tipped balance. Fourteen Points (Wilson's peace plan) — League of Nations. Treaty of Versailles — harsh on Germany. US never joined League — Congress rejected Treaty.
🎭 Roaring 20s & Great Depression: 1920s — economic boom, consumer culture (cars — Ford Model T), radio, movies, Jazz Age, Harlem Renaissance (Langston Hughes, Zora Neale Hurston). Red Scare (fear of communism) — Palmer Raids, Sacco & Vanzetti. Immigration Quotas (1924) — restricted Southern/Eastern Europeans, banned Asians. Scopes Trial (evolution vs. creationism). Stock market crash (1929) → Great Depression — 25% unemployment, Dust Bowl (Okies). New Deal (FDR) — relief, recovery, reform. AAA, CCC, WPA, Social Security. Critics: Liberty League (too much gov), Huey Long, Father Coughlin. Did New Deal solve Depression? Partially — WWII spending finally ended it.
🌐 WWII (1939-1945): US neutral initially (Neutrality Acts), but Lend-Lease aid to Allies. Pearl Harbor (Dec 7, 1941) — US enters. War in Europe: D-Day (June 1944), Battle of the Bulge, fall of Berlin. War in Pacific: island-hopping, Midway (turning point), Iwo Jima, Okinawa. Atomic bombs (Hiroshima & Nagasaki — Aug 1945) — Japan surrendered. Home front: propaganda, rationing, women in workforce (Rosie the Riveter), Japanese Internment (Korematsu v. US). Double V Campaign — Black Americans fighting for democracy abroad while facing discrimination at home.
🎥 Heimler's Period 7 Review❄️ Cold War Begins (1945-1960): US vs. USSR — ideological struggle (democracy/capitalism vs. communism). Containment (Truman Doctrine) — stop spread of communism. Marshall Plan — rebuilt Europe. Berlin Blockade/Airlift (1948). NATO vs. Warsaw Pact. Korean War (1950-53) — US fought communist North Korea/China — stalemate. HUAC & McCarthyism — anti-communist hysteria, blacklists, ruined innocent lives. Arms race — nuclear weapons, Sputnik (1957) → NASA + education reforms. Eisenhower's Farewell Address — warned of "military-industrial complex."
✊ Civil Rights Movement (1954-1968): Brown v. Board of Education (1954) — overturned "separate but equal." Montgomery Bus Boycott (1955) — Rosa Parks, MLK Jr. (nonviolent resistance). Little Rock Nine (1957) — federal troops enforced desegregation. Sit-ins, Freedom Rides, March on Washington (1963) — "I Have a Dream." Civil Rights Act (1964) — banned segregation in public accommodations. Voting Rights Act (1965) — banned literacy tests. Meanwhile, Malcolm X and Black Power (Stokely Carmichael) argued for self-defense/separation. Assassinations: JFK (1963), Malcolm X (1965), MLK (1968), RFK (1968).
📺 1960s — Kennedy & Johnson: JFK's New Frontier — space program, Peace Corps. Cuban Missile Crisis (1962) — closest to nuclear war. LBJ's Great Society — War on Poverty, Medicare/Medicaid, Head Start, immigration reform (1965). Vietnam War — Gulf of Tonkin Resolution (1964) escalated US involvement. Tet Offensive (1968) — turned US public against war. Anti-war movement — college protests, Kent State shooting (1970).
🌅 1970s — Nixon to Carter: Nixon — détente with USSR/China, moon landing (1969), Watergate scandal (1972-74) — Nixon resigned. War Powers Act (1973) — limited presidential war-making. Oil crisis (1973) — OPEC embargo, stagflation (high inflation + unemployment). Carter — human rights focus, Panama Canal Treaty, Camp David Accords. Iran Hostage Crisis (1979-81) — crippled Carter's presidency. Rising conservatism.
🌟 Other Social Movements: Women's rights — feminism's "second wave," Title IX, Roe v. Wade (1973), ERA (failed). Chicano Movement (Cesar Chavez — United Farm Workers). American Indian Movement (AIM) — Wounded Knee occupation (1973). Gay rights — Stonewall Riots (1969). Environmentalism — Silent Spring (Rachel Carson), first Earth Day (1970), EPA created.
🎥 Heimler's Period 8 Review🔵 Reagan Revolution (1980-1988): "Government is not the solution — government is the problem." Reaganomics — supply-side economics: tax cuts (especially for wealthy), deregulation, reduced social spending. Deficit grew massively (military spending). Strong anti-communist stance — "Mr. Gorbachev, tear down this wall!" Iran-Contra Affair — scandal: sold arms to Iran, funded Nicaraguan Contras. End of Cold War — Berlin Wall fell (1989), USSR dissolved (1991).
🟢 1990s — Clinton & New Democrats: NAFTA (1994) — free trade with Canada/Mexico — debates over job loss vs. economic growth. Welfare reform (1996) — ended federal entitlement, gave block grants to states. Tech boom — internet, dot-com bubble. Impeachment of Clinton (1998) — affairs with Monica Lewinsky — acquitted by Senate. Post-Cold War foreign policy: Somalia ("Black Hawk Down"), Bosnia/Kosovo intervention.
🔴 2000s — Bush & New Challenges: 2000 election — Bush v. Gore (Supreme Court stopped recount). 9/11 (2001) — terrorist attacks by Al-Qaeda. War on Terror — Afghanistan (Taliban), Iraq War (2003) — WMDs not found. Patriot Act — expanded surveillance. Great Recession (2008) — housing bubble burst, banks collapsed, TARP bailout.
🟣 2010s-Present — Polarization & Change: Obama (2008) — first Black president. Affordable Care Act (2010) — expanded healthcare. Legalization of same-sex marriage (Obergefell v. Hodges, 2015). Rise of social media and its impact on politics. Tea Party — conservative backlash to Obama. Trump presidency (2016-2020) — populism, immigration crackdown, tax cuts, three Supreme Court justices (ending Roe v. Wade in 2022). COVID-19 pandemic (2020) — economic shutdown, stimulus, vaccines. Increased political polarization. Black Lives Matter — George Floyd protests (2020). Biden presidency (2021-). Changing demographics — US becoming more diverse.
🔮 Key Trends to Know: Globalization debates (winners/losers), technology/AI revolution, climate change policy debates, healthcare reform debates, demographic shifts (aging population, growing Latino/Asian communities), debates over America's role in the world.
🎥 Heimler's Period 9 ReviewCh 1 — Functions: A function is a relation where each input maps to exactly one output (vertical line test). Domain = all valid inputs (avoid division by zero, negative under square roots). Range = all possible outputs. Represented: tables, graphs, equations, function notation f(x). Key operations: composition f(g(x)), transformations (shifts, stretches, reflections), and finding inverse functions (f⁻¹(f(x)) = x). A function is one-to-one if it passes the horizontal line test.
Ch 2 — Linear Functions: f(x) = mx + b. Slope m = Δy/Δx represents rate of change. Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes. Linear models: direct variation (y = kx), scatter plots, least-squares regression.
Ch 3 — Polynomial & Rational: Complex numbers (a + bi where i = √(−1)). Quadratic functions f(x) = ax² + bx + c — vertex form: f(x) = a(x − h)² + k. Power functions f(x) = kxⁿ. Higher-degree polynomials: degree determines max turns, leading coefficient determines end behavior. Zeros → factors: if f(c) = 0 then (x − c) is a factor. Use polynomial long division and synthetic division. Rational functions: f(x) = p(x)/q(x). Vertical asymptotes where denominator = 0 (excluding holes). Horizontal asymptotes determined by degree comparison. Slant asymptotes when numerator degree = denominator degree + 1.
Ch 4 — Exponential Functions: f(x) = abˣ with base b > 0, b ≠ 1. If b > 1 → growth (increasing). If 0 < b < 1 → decay (decreasing). Natural base e ≈ 2.718 — used for continuous growth/decay models (compound interest A = Peʳᵗ, population N = N₀eᵏᵗ). Graph features: y-intercept at (0,1), horizontal asymptote y = 0, domain (−∞,∞), range (0,∞).
Logarithmic Functions: The inverse of exponential functions. y = log_b(x) means bʸ = x. Common log (base 10): log(x). Natural log (base e): ln(x). Graph: vertical asymptote at x = 0, domain (0,∞), range (−∞,∞). Key properties: log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) − log_b(N), log_b(Mⁿ) = n·log_b(M). Change of base: log_b(M) = log(M)/log(b).
Solving Equations: Exponential → isolate the exponential term, take log of both sides. Logarithmic → rewrite in exponential form, use one-to-one property. Always check for extraneous solutions (log arguments must be positive!). Modeling: exponential growth/decay, half-life, logistic growth, Newton's Law of Cooling, Richter scale (logarithmic), pH scale.
Ch 5 — Angles & Trig Functions: Angles in standard position (vertex at origin, initial side on +x-axis). Radians: 2π rad = 360°, conversion: θ/180 = θ_R/π. Coterminal angles: add/subtract 360° or 2π. Arc length: s = rθ. Unit circle: (cos t, sin t) at angle t radians. Special angles: 0, π/6, π/4, π/3, π/2. Right triangle trig: SOH-CAH-TOA. Pythagorean identity: sin²θ + cos²θ = 1.
Ch 6 — Periodic Functions: Graphs of sin/cos: amplitude = |a|, period = 2π/|b|, phase shift = −c/b, midline = d for y = a·sin(bx + c) + d. Tan graph: period = π, vertical asymptotes at cos x = 0. Inverse trig: arcsin (range [−π/2, π/2]), arccos (range [0, π]), arctan (range (−π/2, π/2)).
Ch 7 — Identities & Equations: Fundamental identities: reciprocal, quotient, Pythagorean (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ). Sum/difference formulas, double-angle (sin 2θ = 2 sin θ cos θ), half-angle. Solving trig equations: isolate the trig function, use inverse trig, consider all solutions within [0, 2π) by adding multiples of the period.
Ch 8 — Polar & Vectors: Polar coordinates: (r, θ) where x = r cos θ, y = r sin θ, r = √(x² + y²). Polar graphs: cardioids (r = a ± a cos θ), roses (r = a cos(nθ)), limacons. Vectors: magnitude |v|, direction, addition (head-to-tail), scalar multiplication. Complex numbers in polar form: z = r(cos θ + i sin θ). De Moivre's Theorem: zⁿ = rⁿ(cos nθ + i sin nθ).
Ch 9 — Systems of Equations: Solve linear systems: substitution, elimination, Gaussian elimination with matrices. Matrices: addition, scalar multiplication, multiplication (A·B valid when A's columns = B's rows). Determinant of 2×2: |A| = ad − bc. Inverse A⁻¹ exists if det ≠ 0. Solve systems using inverse: X = A⁻¹B. Cramer's Rule.
Ch 10 — Analytic Geometry: Conic sections: parabolas (focus/directrix), ellipses (sum of distances constant), hyperbolas (difference of distances constant), circles (special ellipse). Parametric equations: x = f(t), y = g(t) — model motion and curves.
Ch 11 — Sequences & Probability: Arithmetic sequences: aₙ = a₁ + (n−1)d. Geometric sequences: aₙ = a₁rⁿ⁻¹. Series: sum of sequence terms. Binomial Theorem: (x + y)ⁿ = Σ C(n,k) xⁿ⁻ᵏ yᵏ. Probability: P(event) = favorable/total outcomes.
Ch 12 — Introduction to Calculus: Limits: as x approaches a, f(x) approaches L. One-sided limits, two-sided limits exist when both sides agree. Continuity: a function is continuous if lim_{x→a} f(x) = f(a). The derivative as a limit of difference quotients (instantaneous rate of change).
📌 Note: Unit 4 NOT on the AP exam. Focus Units 1-3 for May!