Algebra II · Units 1–9 · Quick reference

College Entrance Math Review(SAT · ACT · PSAT)

Grades 9–128 pages

Algebra, advanced math, problem solving and data analysis, and geometry and trigonometry, with the three test formats side by side, strategies and original practice. Not affiliated with or endorsed by any test maker. SAT, PSAT, CLEP and ACT are trademarks of their owners.

1The three tests2Strategies3Numbers4Linear5Systems6Functions7Function families8Quadratics9Exponents & polynomials10Polynomial graphs11Exponentials & logs12Ratios & percents13Word problems14Statistics15Probability & inference16Angles & triangles17Area, volume, circles18Trigonometry19Number & quantity20Formulas21Practice

1The Three Tests at a Glanceconfirm details before test day

Formats, from public information (Oct 2026)
Math section	SAT	PSAT/NMSQT · PSAT 10 · PSAT 8/9	ACT
Questions	44 (40 scored + 4 unscored pretest)	44 (40 scored + 4 unscored pretest)	45 (41 scored)
Time and pace	70 min: two 35-min modules (≈ 1.6 min each)	same as the SAT	50 min (≈ 1.1 min each)
Format	digital, adaptive (module 2 depends on module 1); easy → hard in each module	same as the SAT	paper or online; not adaptive
Answer types	multiple choice (4 choices) + typed answers (about 1 in 4)	same as the SAT	multiple choice, 4 choices
Calculator	whole section: built-in graphing calculator or your own approved one	same as the SAT	your own 4-function, scientific or graphing (no CAS); built in online
Formulas	reference sheet provided	reference sheet provided	none given: memorize them
Math score	200–800	160–760; PSAT 8/9: 120–720	1–36

What is tested (approximate share of scored math questions)
SAT and PSAT domain	Share	See	ACT reporting category	Share	See
Algebra (linear)	≈ 35 %	§4–5	Number & Quantity	10–12 %	§3, §9, §19
Advanced Math (nonlinear)	≈ 35 %	§6–11	Algebra; Functions	17–20 % each	§4–11
Problem Solving & Data Analysis	≈ 15 %	§12–15	Geometry (with trig)	17–20 %	§16–18
Geometry & Trigonometry	≈ 15 %	§16–18	Statistics & Probability	12–15 %	§14–15
(PSAT tests: same four domains)			Integrating Essential Skills	≈ 20 %	§3, §12–13

CLEP College Algebra (college credit)

About 60 questions in 90 minutes, some unscored.
Built-in scientific (non-graphing) calculator only.
Scored 20–80; ACE recommends credit at 50 (3 semester hours); colleges set their own policy.
Study §4–11 and §19.

Scoring facts

No wrong-answer penalty on any of these tests: never leave a blank.
SAT and PSAT modules run easy → hard: bank the early points.
Practice with the free full-length tests each test maker posts.

2Strategies That Workplug in, backsolve, graph it

Plug in numbers

When the answer choices are expressions, pick an easy value (x = 5; avoid 0 and 1), work out the target number, then test each choice.

EXWhich equals x² − 9x − 3 (x ≠ 3)? Try x = 5: 162 = 8. Only x + 3 gives 8.

Reread the last line

Before you choose, read the question’s last line again: it may want 3x, the units or the total, not x.

Backsolve

When the choices are numbers, start with a middle choice, test it in the problem, then go up or down.

EXA number plus 40 % of itself is 84. Try 60: 60 + 24 = 84 ✓, so the number is 60.

Use the graphing calculator

Equations, systems: graph each side (or each equation); read where they cross.
Quadratics: tap the graph for the vertex and zeros.
Data: enter a table and fit a line or curve.

Student-produced responses (SAT, PSAT)

Type a fraction or a decimal: up to 5 characters, 6 with a minus sign.
No mixed numbers (type 7/2, not 3 1/2), and no %, $ or commas.
If several answers work, enter one.

Common traps

Watch units and the words not, except and least.
Check for extraneous roots after squaring or clearing fractions.

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

3Numbers & Operationsthe arithmetic every test assumes

Kinds of numbers
Set	Members	Example
Integers	…, −2, −1, 0, 1, 2, …	−7, 0, 12
Rational	a ratio of integers pq (q ≠ 0)	34, −2.5, 0.3̄
Irrational	not a ratio: decimal never ends or repeats	√2, π
Prime	exactly two factors, 1 and itself	2, 3, 5, 7, 11, 13

2 is the only even prime; 1 is not prime. Even × anything = even; odd × odd = odd.

Factors and multiples Worked

GCF(36, 60) = 12; LCM(36, 60) = 180GCF × LCM = 36 × 60
36 = 2² · 3²; 60 = 2² · 3 · 5prime factorizations
47 ÷ 6 = 7 remainder 547 = 6 · 7 + 5

Fractions and order of operations Worked

+ 58 = 16 + 1524 = 3124common denominator
÷ 910 = 34 × 109 = 56flip and multiply
− 3(4 − 6)² ÷ 2 = 20 − 3 · 4 ÷ 2 = 14parentheses, powers, × ÷, + −

Scientific notation

a × 10ⁿ with 1 ≤ a < 10.
Multiply: multiply the a’s, add the exponents.

EX(3 × 10⁴)(6 × 10⁻⁷) = 18 × 10⁻³ = 1.8 × 10⁻²; 0.00042 = 4.2 × 10⁻⁴.

Number sense

Absolute value: |x| is the distance from 0: |−8| = 8; |x − 3| ≤ 2 means 1 ≤ x ≤ 5
Mean of a group: total ÷ count; the total = mean × count
Digits: 3,482.6: 8 is in the tens place, 6 in the tenths

4Linear Equations & Inequalitiesslope, intercepts, forms

Forms of a line
Form	Equation	Shows
Slope-intercept	y = mx + b	slope m, y-intercept b
Point-slope	y − y₁ = m(x − x₁)	slope and one point
Standard	Ax + By = C	slope −AB; intercepts CA, CB
Horizontal; vertical	y = k; x = h	slope 0; undefined

Slope

(−1, 4) and (3, −4): m = −84 = −2, so y = −2x + 2 (x-intercept 1).

m = y₂ − y₁x₂ − x₁

Down 2 for every 1 to the right: slope −2.

Parallel and perpendicular

Parallel lines: equal slopes.
Perpendicular lines: slopes are negative reciprocals (product −1).

EXThrough (2, 5), perpendicular to y = 23x + 1: m = −32, so y = −32x + 8.

Solve Worked

2x − 13 = x + 42 → 4x − 2 = 3x + 12 → x = 14cross-multiply
−2(x − 5) ≥ 14 → x − 5 ≤ −7 → x ≤ −2dividing by −2 flips ≥
|2x − 3| = 7 → 2x − 3 = 7 or −7 → x = 5 or −2two cases

Read the meaning

Linear model: C = 45 + 30h: 45 is the starting fee (h = 0); 30 is the added cost per hour (the slope)
How many solutions?: ax + b = cx + d has one if a ≠ c, none if a = c and b ≠ d, infinitely many if a = c and b = d

5Systems & Linear Modelstwo equations, two unknowns

Elimination Worked

3x + 2y = 16 and x − 2y = 0

add: 4x = 16 → x = 4, then y = 2the y-terms cancel

Substitution Worked

y = 2x − 1 and 3x + y = 19

3x + 2x − 1 = 19 → x = 4, y = 7replace y

0 ≤ 2 is true, so shade the side with (0, 0).

The solution is where the lines cross.

Set up a word problem Worked

Adult tickets $12, student $7; 50 tickets brought in $460
a + s = 50 and 12a + 7s = 460one equation per fact
12a + 7(50 − a) = 460 → 5a = 110 → a = 22, s = 28

How many solutions?
Lines	Solutions
different slopes	exactly one
same slope, different intercepts (parallel)	none
same line (one equation is a multiple of the other)	infinitely many

2x + 3y = 6 and 4x + ky = 5 have no solution when k = 6: same slope, but 4x + 6y = 12 ≠ 5.

Inequalities in two variables

Solid boundary for ≤ or ≥, dashed for < or >.
Test (0, 0): if it works, shade its side.
A system’s solutions are where the shadings overlap.

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

6Functionsnotation, graphs, transformations

Function language

Domain; range: all allowed inputs; all outputs
Zero; y-intercept: an x with f(x) = 0; the value f(0)
Average rate of change: f(b) − f(a)b − a: for x² from 1 to 4, 16 − 13 = 5

Evaluate and compose Worked

f(x) = x² − 3x, g(x) = 2x + 1
f(−2) = 4 + 6 = 10
f(g(1)) = f(3) = 0; g(f(1)) = g(−2) = −3inside first

Transformations
Change	Effect on the graph of y = f(x)
f(x) + k	up k (down if k < 0)
f(x − h)	right h (f(x + h): left h)
−f(x); f(−x)	reflect over the x-axis; over the y-axis
a·f(x)	vertical stretch (a > 1) or shrink (0 < a < 1)
f(bx)	horizontal shrink by a factor of b (b > 1)

Dashed y = x²; solid y = (x + 2)² − 5: left 2, down 5, so the vertex moves from (0, 0) to (−2, −5).

Inverse Worked

f(x) = 3x − 6: swap x and y, solve: f⁻¹(x) = x + 63

check: f(4) = 6 and f⁻¹(6) = 4 ✓

7Function Families & Graphsknow the shapes on sight

Domain and range
Family	Domain	Range	Key feature
y = x, y = x³	all reals	all reals	through (0, 0)
y = x²; y = ∣x∣	all reals	y ≥ 0	vertex (0, 0)
y = √x	x ≥ 0	y ≥ 0	starts at (0, 0)
y = 1x	x ≠ 0	y ≠ 0	asymptotes x = 0, y = 0
y = 2ˣ	all reals	y > 0	asymptote y = 0; (0, 1)
y = log₂ x	x > 0	all reals	asymptote x = 0; (1, 0)

Reading a graph

f(a) is the height of the graph at x = a.
Solutions of f(x) = k: the x-values where the graph meets the line y = k.
f(x) > 0 where the graph is above the x-axis.
Number of real zeros = number of x-intercepts.

More function terms

Vertical line test: a graph is a function if no vertical line meets it twice
Piecewise: different rules on different intervals: f(x) = x + 1 for x < 2, 7 − x for x ≥ 2 gives f(1) = 2, f(4) = 3
Increasing: rises left to right; decreasing falls

Domain from a formula Worked

g(x) = 1x + 1: x ≠ −1no zero denominator
h(x) = √(x − 4): x ≥ 4no negative under √
k(x) = ln(x − 2): x > 2log of a positive

8Quadraticsthree forms, roots, vertex

Three forms
Form	Equation	Shows
Standard	ax² + bx + c	y-intercept c; axis x = −b2a
Vertex	a(x − h)² + k	vertex (h, k)
Factored	a(x − r)(x − s)	zeros r and s; axis halfway

a > 0 opens up (minimum); a < 0 opens down (maximum).

Vertex and tangency Worked

y = −2x² + 8x + 1: x = −8−4 = 2, y = 9 (maximum)

y = 2x + k touches y = x² once when x² − 2x − k = 0 has 4 + 4k = 0: k = −1discriminant = 0

Quadratic formula

Discriminant b² − 4ac: > 0 two real roots, = 0 one (double) root, < 0 no real roots (two complex).

x = −b ± √(b² − 4ac)2a

Solve three ways Worked

2x² − 3x − 5 = 0 → (2x − 5)(x + 1) = 0 → x = 52 or −1factor
x² + 6x + 2 = 0 → (x + 3)² = 7 → x = −3 ± √7complete the square
x² − 4x + 1 = 0 → x = 4 ± √122 = 2 ± √3formula

y = x² − 2x − 3: zeros −1 and 3; vertex (1, −4) on the axis x = 1; y-intercept −3.

Sum and product of roots

Sum = −ba; product = ca.

EX2x² − 3x − 5: sum 32, product −52 (check: 52 + (−1), 52 × (−1)).

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

9Exponents, Polynomials & Radicalsequivalent expressions and equations

Exponents and radicals
Rule	Example	Rule	Example
aᵐ · aⁿ = aᵐ⁺ⁿ	x³ · x⁴ = x⁷	a⁰ = 1; a⁻ⁿ = 1aⁿ	5⁻² = 125
aᵐaⁿ = aᵐ⁻ⁿ	x⁹x³ = x⁶	a^m/n = (ⁿ√a)ᵐ	8^2/3 = 4; 27^−1/3 = 13
(aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ	(2x³)² = 4x⁶	√(ab) = √a · √b	√72 = 6√2

Factoring patterns
Pattern	Factored
a² − b²	(a + b)(a − b)
a² + 2ab + b²; a² − 2ab + b²	(a + b)²; (a − b)²
a³ + b³; a³ − b³	(a + b)(a² − ab + b²); (a − b)(a² + ab + b²)
x³ + 2x² − 9x − 18 (grouping)	(x + 2)(x + 3)(x − 3)

Remainder and factor theorems

p(x) ÷ (x − a) leaves remainder p(a).
(x − a) is a factor exactly when p(a) = 0.

EXp(x) = x³ − 4x² + x + 6: p(2) = 0, so p(x) = (x − 2)(x² − 2x − 3) = (x − 2)(x − 3)(x + 1).

Radical equation Worked

√(x + 7) = x + 1 → x + 7 = x² + 2x + 1 → x² + x − 6 = 0square both sides

x = 2 or −3; x = −3 gives √4 = −2 ✗, so x = 2check each root

Rational expressions Worked

x² − 9x² + x − 6 = (x − 3)(x + 3)(x + 3)(x − 2) = x − 3x − 2x ≠ −3, 2

x²x − 2 = 4x − 2 → x² = 4 → x = ±2; x = 2 divides by 0, so x = −2extraneous root

Check in the original

Squaring both sides or clearing denominators can add roots that do not work.

10Polynomial & Rational Graphszeros, end behavior, asymptotes

y = (x + 1)(x − 2)² crosses at −1 (single zero) and touches at 2 (double zero); y-intercept 4.

End behavior
Leading term	Left end	Right end
even degree, a > 0	up	up
even degree, a < 0	down	down
odd degree, a > 0	down	up
odd degree, a < 0	up	down

Zeros and multiplicity

A factor (x − r) gives a zero at x = r.
Odd multiplicity: the graph crosses. Even multiplicity: it touches and turns back.

y = x + 1x − 2: vertical asymptote x = 2, horizontal asymptote y = 1, zero at −1.

Rational functions

Vertical asymptote: a zero of the denominator that does not cancel
Hole: a factor that cancels: (x − 3)(x + 1)x − 3 has a hole at x = 3
Horizontal asymptote: degree top < bottom: y = 0; equal: y = ratio of leading coefficients; top > bottom: none

11Exponential Models & Logarithmsgrowth, decay, logs

Exponential model

a = starting amount; b > 1 grows by r = b − 1 per step; 0 < b < 1 decays.

y = a · bˣ = a(1 ± r)ᵗ

Models Worked

$2,000 at 5 % a year for 3 years: 2,000(1.05)³ = $2,315.25
600 mg, losing 20 % per hour, after 2 h: 600(0.8)² = 384 mg
50 cells doubling every 4 h: 50 · 2t/4; at t = 12, 400

Linear or exponential?

Linear: adds the same amount each step: 5, 8, 11, 14 (differences 3)
Exponential: multiplies by the same factor: 3, 6, 12, 24 (ratios 2)

Compound interest

Second form: continuous. $1,000 at 6 % monthly for 2 years: 1,000(1.005)²⁴ ≈ $1,127.16.

A = P(1 + rn)ⁿᵗ

A = Peʳᵗ

Logarithms (ACT, CLEP)
Log rule	Example
log_b x = y means bʸ = x	log₂ 32 = 5; log₃ 19 = −2
log(xy) = log x + log y	log 4 + log 25 = log 100 = 2
log(x/y) = log x − log y	log₂ 48 − log₂ 3 = log₂ 16 = 4
log(xⁿ) = n log x	log 10⁵ = 5
log_b x = ln xln b	3ˣ = 20 → x = ln 20ln 3 ≈ 2.727

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

12Ratios, Rates, Percents & Unitsproportional reasoning

Percent

part = percent × whole

% change = new − oldold × 100

Percent problems Worked

$80 jacket, 25 % off, then 8 % tax: 80(0.75)(1.08) = $64.80
Up 20 %, then down 20 %: 1.2 × 0.8 = 0.96 → 4 % lowernot back to even
After a 15 % raise the pay is $27.60/h; before: 27.60 ÷ 1.15 = $24.00divide by the multiplier

Percent shortcuts

10 %: move the decimal point one place left; 5 % is half of that.

EX15 % tip on $42: 4.20 + 2.10 = $6.30.

Multiply by the multiplier

Up 7 % means × 1.07; down 7 % means × 0.93. Chain several changes by multiplying the multipliers.

Ratios and variation

a : b = 3 : 5 means a = 3k and b = 5k (total 8k).
Direct variation y = kx; inverse variation y = kx.

EXSplit 96 in a 3 : 5 ratio: k = 12, so 36 and 60. Inverse: y = 12 when x = 3 gives k = 36, so y = 4 at x = 9.

Units Worked

mi/h = 60 × 5,280 ft3,600 s = 88 ft/scancel units
ft × 4 ft = 36 in × 48 in = 1,728 in²1 ft² = 144 in²
g in 100 cm³: density 2.5 g/cm³

Equivalents to know
Fraction	Decimal	Percent
18; 15	0.125; 0.2	12.5 %; 20 %
14; 13	0.25; 0.3̄	25 %; 3313 %
12; 23	0.5; 0.6̄	50 %; 6623 %
34; 32	0.75; 1.5	75 %; 150 %

Proportions and rates Worked

3 cups of flour for 24 cookies; for 60: 3 × 6024 = 7.5 cups
18 oz for $4.50 ($0.25/oz) or 24 oz for $5.76 ($0.24/oz): 24 oz is cheaper per ounceunit price
map 1 in : 15 mi; 3.2 in = 48 mi

Time and rate conversions Worked

$3.20 per pound = 3.20 ÷ 16 = $0.20 per ounce

2.25 h = 2 h 15 min; 135 min = 2.25 h0.25 h = 15 min

13Word Problems & Modelingturn words into equations

Translate
Words	Math	Words	Math
is, equals	=	of (with a percent)	×
more than, increased by	+	per, for each	÷ (a rate)
less than, fewer	−	at least; at most	≥; ≤
5 less than x	x − 5	twice the sum of x and 3	2(x + 3)

Distance, rate, time

Two cars leave the same town in opposite directions at 50 and 60 mi/h: they are 330 mi apart after 330 ÷ 110 = 3 h.

d = rt

Mixture Worked

How many liters of 40 % acid with 6 L of 10 % acid make a 20 % mix?

0.40x + 0.10(6) = 0.20(x + 6) → 0.2x = 0.6 → x = 3 Lacid in = acid out

Money Worked

Simple interest I = Prt: $1,500 at 4 % for 3 years = $180
6 % commission on a $240,000 sale = $14,400
$18 after a 25 % markup: cost = 18 ÷ 1.25 = $14.40

Work together Worked

A fills a tank in 3 h, B in 6 h: 13 + 16 = 12 tank per hour → 2 hadd the rates

Integers and ages Worked

Three consecutive even integers sum to 78: 3n + 6 = 78 → 24, 26, 28n, n + 2, n + 4

Ana is 3 times Ben’s age; in 8 years she will be twice his: 3b + 8 = 2(b + 8) → Ben 8, Ana 24

Interpret the pieces

In y = mx + b: m is the rate, b the starting value.
In y = a(b)ˣ: a is the start, b the growth factor per period.
In h = −16t² + vt + h₀: v is the launch speed, h₀ the launch height.

Build a model Worked

A plan costs $25 plus $0.10 per text: C = 25 + 0.10t; at 340 texts, C = $59linear
A town of 8,000 grows 3 % a year: P = 8,000(1.03)ᵗ; after 10 years ≈ 10,751exponential
A ball thrown up at 48 ft/s from 5 ft: h = −16t² + 48t + 5; highest at t = 1.5 s, h = 41 ftquadratic

h = −16t² + 48t + 5: the vertex (1.5, 41) is the highest point; the graph starts at 5 ft.

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

14Statistics & Datacenter, spread, displays

Center and spread

Mean; median: sum ÷ count; middle value in order (mean of the two middles if even)
Mode; range: most frequent value; largest − smallest
Standard deviation: typical distance from the mean: bigger means more spread
IQR; outlier: Q3 − Q1; a value beyond 1.5 × IQR from Q1 or Q3

Changing every value

Add c: mean and median go up by c; range and SD stay the same.
Multiply by c: mean, median, range and SD are all multiplied by c.

One data set Worked

2, 4, 4, 5, 7, 9, 11: mean 6, median 5, mode 4, range 9
add 30: mean 728 = 9, median 5 + 72 = 6an outlier pulls the mean
mean of 5 scores is 85; four are 80, 92, 75, 88 → fifth = 425 − 335 = 90total = mean × count

Displays
Display	Read it for
Histogram; dot plot	shape (symmetric, skewed), center, gaps
Box plot	min, Q1, median, Q3, max; IQR = box width
Scatterplot	direction, form, strength, outliers
Two-way table	counts by two categories (§15)

Strong positive linear association. Slope 4.6: about 4.6 more points per hour. At x = 2 the line predicts 67.3, so the point (2, 70) has residual 70 − 67.3 = 2.7.

Q1 = 6, median 8, Q3 = 11, IQR 5; fences −1.5 and 18.5, so 24 is an outlier.

15Probability & Inferencetwo-way tables, samples, studies

A two-way table (original data)
Lunch choice	Hot	Packed	Total
Grade 9	42	18	60
Grade 10	30	30	60
Total	72	48	120

Read it Worked

P(hot) = 72120 = 0.6whole table
P(hot ∣ grade 10) = 3060 = 0.5grade 10 row only
P(grade 10 ∣ packed) = 3048 = 0.625packed column only

Probability rules

P(not A) = 1 − P(A).
Independent events: P(A and B) = P(A) · P(B).
P(A or B) = P(A) + P(B) − P(A and B).

EXTwo heads and then a 6: 14 × 16 = 124.

Estimate from a sample Worked

58 % ± 4 % → between 54 % and 62 %

46 of 200 sampled prefer A; of 5,000 people, expect 5,000 × 46200 = 1,150same proportion

Sampling and studies

Random sample: results can be generalized to the population sampled
Random assignment: needed to conclude cause and effect
Observational study: shows association, not cause
Margin of error: estimate ± MOE is the plausible range; larger samples give smaller margins

16Lines, Angles & Trianglesangles, similarity, Pythagoras

Angle facts

Vertical angles are equal; a linear pair sums to 180°.
Parallel lines and a transversal: corresponding and alternate interior angles are equal; same-side interior angles sum to 180°.
Triangle angles sum to 180°; an exterior angle equals the two remote interior angles together.
Polygon with n sides: interior angles sum to (n − 2) · 180° (hexagon 720°).

The two special right triangles.

Similar triangles Worked

Corresponding sides are proportional; angles are equal
a 6-ft post casts a 4-ft shadow; a tree casts 18 ft: 64 = h18 → h = 27 ft
scale factor k: lengths × k, areas × k², volumes × k³k = 3: area × 9

Special right triangles Worked

30-60-90, hypotenuse 10: legs 5 and 5√3 ≈ 8.66
45-45-90, leg 7: hypotenuse 7√2 ≈ 9.90
square with diagonal 12: side 12√2 = 6√2

Triangle toolkit
Fact	Detail
Pythagorean theorem	a² + b² = c² (c is the hypotenuse)
Common triples	3-4-5, 5-12-13, 8-15-17, 7-24-25 and multiples (6-8-10)
Triangle inequality	each side < the sum of the other two: sides 4 and 9 → third between 5 and 13
Isosceles	equal sides face equal angles

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

17Area, Volume & Circlesformulas and circle equations

Area and volume
Figure	Formula	Figure	Formula	Solid	Volume
Rectangle	A = lw	Circle	A = πr²; C = 2πr	Box	V = lwh
Triangle	A = 12bh	Sector	A = θ360° · πr²	Cylinder	V = πr²h
Parallelogram	A = bh	Arc length	s = θ360° · 2πr = rθ (radians)	Cone; pyramid	V = 13πr²h; 13Bh
Trapezoid	A = 12(b₁ + b₂)h	Rectangular box	surface 2(lw + lh + wh)	Sphere	V = 43πr³; S = 4πr²

Circle equation

center (h, k), radius r. Complete the square: x² + y² − 6x + 4y − 12 = 0 → (x − 3)² + (y + 2)² = 25: center (3, −2), r = 5.

(x − h)² + (y − k)² = r²

A central angle equals its arc; an inscribed angle is half of it (100° and 50° here), so one on a diameter is 90°. A tangent is perpendicular to the radius.

Worked Worked

r = 6, θ = 120°: arc = 4π ≈ 12.57; sector = 12π ≈ 37.70
cylinder r = 3, h = 8: 72π ≈ 226.19; sphere r = 3: 36π ≈ 113.10
(−2, 1) to (4, 9): distance √(36 + 64) = 10; midpoint (1, 5)d = √((Δx)² + (Δy)²)

18TrigonometrySOH-CAH-TOA, radians, unit circle

Label the sides from θ’s point of view.

Right-triangle ratios

sin θ = oppositehypotenuse, cos θ = adjacenthypotenuse, tan θ = oppositeadjacent

EXIn the figure: sin θ = 35, cos θ = 45, tan θ = 34.

SOH-CAH-TOA (Traditional)

Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.

Cofunctions and identity

sin x° = cos(90° − x°): sin 28° = cos 62°.
sin²θ + cos²θ = 1.

Radians

135° = 3π4; 5π6 = 150°. Unit circle: the point at angle θ is (cos θ, sin θ).

π radians = 180°

Exact values
θ	0°	30°	45°	60°	90°
sin θ	0	12	√22	√32	1
cos θ	1	√32	√22	12	0
tan θ	0	√33	1	√3	undefined

ACT extras

Law of sines: asin A = bsin B = csin C
Law of cosines: c² = a² + b² − 2ab cos C
y = A sin(Bx) + D: amplitude ∣A∣, period 2π∣B∣, midline y = D: y = 3 sin 2x + 1 has amplitude 3, period π, range −2 to 4

19Number & Quantity (ACT, CLEP)beyond the SAT list

Complex numbers Worked

i = √−1, i² = −1; powers repeat every 4: i²³ = i³ = −i
(3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 11 − 10i
5 + 5i1 + 2i · 1 − 2i1 − 2i = 15 − 5i5 = 3 − imultiply by the conjugate

Sequences
	Arithmetic (add d)	Geometric (multiply by r)
nth term	aₙ = a₁ + (n − 1)d	aₙ = a₁rⁿ⁻¹
Sum of n terms	Sₙ = n(a₁ + aₙ)2	Sₙ = a₁(1 − rⁿ)1 − r
Example	7, 11, 15, …: a₂₀ = 83; S₂₀ = 900	3, 6, 12, …: a₈ = 384

Matrix work Worked

det of rows (2, 1) and (3, 4) = 2 · 4 − 1 · 3 = 5

rows (1, 2), (0, 1) times column (3, 4) = column (11, 4)row · column

Counting

Permutations (order matters): ₙPᵣ = n!(n − r)!.
Combinations (order does not): ₙCᵣ = n!r!(n − r)!.

EX8 runners, gold-silver-bronze: 8 · 7 · 6 = 336. A 3-person committee from 8: 56.

College Entrance Math ReviewAlgebra II · Units 1–9 · continued

20Formulas to Know Coldthe ACT gives no sheet

Formula summary
Topic	Formula	Topic	Formula	Topic	Formula
Slope	m = y₂ − y₁x₂ − x₁	Quadratic formula	x = −b ± √(b² − 4ac)2a	Circle	(x − h)² + (y − k)² = r²
Line	y = mx + b	Vertex	x = −b2a	Arc; sector	θ360 · 2πr; θ360 · πr²
Distance	√((x₂ − x₁)² + (y₂ − y₁)²)	Growth or decay	y = a(1 ± r)ᵗ	Pythagorean	a² + b² = c²
Midpoint	(x₁ + x₂2, y₁ + y₂2)	Interest	A = P(1 + rn)ⁿᵗ	Special triangles	x, x, x√2; x, x√3, 2x
Percent change	new − oldold × 100	Arithmetic	aₙ = a₁ + (n − 1)d	SOH-CAH-TOA	opphyp, adjhyp, oppadj
Mean	sumcount	Geometric	aₙ = a₁rⁿ⁻¹	Cylinder; cone	πr²h; 13πr²h
Probability	favorabletotal	Combinations	ₙCᵣ = n!r!(n − r)!	Sphere	43πr³; 4πr²

Common mistakes
Mistake	Fix	Mistake	Fix
(a + b)² = a² + b²	(a + b)² = a² + 2ab + b²	√(9 + 16) = 3 + 4	√25 = 5, not 7
−3² = 9	−3² = −9; (−3)² = 9	x² = 25 → x = 5	x = ±5
a + ba = b	a + ba = 1 + ba	up 20 %, down 20 % = no change	1.2 × 0.8 = 0.96: 4 % less
flipping ≤ after × by a positive	flip only for a negative	trig in the wrong mode	check degrees or radians first

21Practice Setoriginal problems, answers checked

Thirty-four practice problems (original; not from any test)
#	Problem	Answer	#	Problem	Answer
1	(Algebra) Solve 5x − 7 = 2x + 11	x = 6	9	(Adv. math) Simplify x² − 16x + 4	x − 4 (plug in x = 1: −3)
2	(Algebra) Line through (1, −2) and (4, 7)	y = 3x − 5	10	(Data) $60, up 10 %, then down 10 %	$59.40
3	(Algebra) x + y = 10 and x − y = 4	(7, 3)	11	(Data) Mean of 6, 9, 10, 15 and x is 11; find x	15
4	(Algebra) For which c does 3x + 6 = 3x + c have infinitely many solutions?	c = 6	12	(Data) From the §15 table, P(packed ∣ grade 9)	0.3
5	(Adv. math) Zeros of x² − 5x − 14	7 and −2	13	(Geometry) Right triangle with legs 9 and 12: hypotenuse	15 (3-4-5 × 3)
6	(Adv. math) Vertex of y = 2(x − 3)² + 5	(3, 5), minimum 5	14	(Geometry) Center and radius of x² + y² + 8x − 2y = 8	(−4, 1), r = 5
7	(Adv. math) f(x) = 4x − 1; find f(f(2))	27	15	(Trig) sin A = 513 in a right triangle; cos A and tan A	1213; 512
8	(Adv. math) Solve 2^{x + 1} = 32	x = 4 (backsolve: 2⁵ = 32)	16	(ACT) (2 − 3i)(2 + 3i)	13

Eighteen more
#	Problem	Answer	#	Problem	Answer
17	(Numbers) GCF and LCM of 18 and 24	6; 72	26	(Geometry) Arc length, r = 9, central angle 40°	2π ≈ 6.28
18	(Numbers) (4 × 10⁵)(5 × 10⁻²) in scientific notation	2 × 10⁴	27	(Trig) Convert 210° to radians	7π6
19	(Algebra) Solve −3(x + 2) < 9	x > −5	28	(ACT) 5th term of 2, 6, 18, …; ₆C₂	162; 15
20	(Algebra) Slope of a line perpendicular to 4x − 2y = 7	−12	29	(Algebra) 3x + 2y = 12 and x − y = −1	(2, 3)
21	(Adv. math) Number of real solutions of x² + 4x + 5 = 0	none (discriminant −4)	30	(Adv. math) Simplify (3x²y)³	27x⁶y³
22	(Adv. math) Solve √(2x + 1) = 5	x = 12	31	(Adv. math) Zeros of y = x(x − 4)²: cross or touch?	0 crosses; 4 touches
23	(Data) A $250 bike is marked down to $200; percent decrease	20 %	32	(Adv. math) Horizontal asymptote of y = 6x + 12x − 5	y = 3
24	(Data) Pipe A fills a pool in 4 h, pipe B in 12 h; together	3 h	33	(Data) Mean and median of 4, 8, 8, 10, 20	10; 8
25	(Geometry) Interior angle sum of an octagon	1,080°	34	(Geometry) Volume of a cone, r = 3, h = 4	12π ≈ 37.70