๐โ๏ธ๐
SAT MATH
Core Topics โ Self-Study NotebookName: _________________________ Date: ______________
๐ SECTION 1 โ Linear Equations & Inequalities
1
LINEAR EQUATIONS
Algebra
โ
โโ
๐ง MEMORY KEY
ISOLATE โ SIMPLIFY โ SOLVE
Whatever you do to one side, do to the other.
๐ EXAMPLE
Solve: \(3(x - 4) = 2x + 1\)
\(3x - 12 = 2x + 1 \;\Rightarrow\; x = 13\)
\(3x - 12 = 2x + 1 \;\Rightarrow\; x = 13\)
Question: If \(5(2x - 3) - 4 = 3(x + 2) + x\), what is the value of \(x\)?
โ ๏ธ
COMMON TRAP: Forgetting to distribute the negative sign โ always expand brackets first!
โ๏ธ My Work & Answer:
2
SYSTEMS OF EQUATIONS
Algebra
โ
โ
โ
๐ง MEMORY KEY
ELIMINATION = add/subtract ยท SUBSTITUTION = plug in
๐ EXAMPLE
\(\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}\)
Add both: \(3x = 9 \Rightarrow x = 3, y = 1\)
Question: In the system below, what is the value of \(x + y\)?
\[\begin{cases} 4x - 3y = 10 \\ 2x + 3y = 14 \end{cases}\]
A) 4
B) 6
C) 8
D) 10
โ ๏ธ
The question asks for \(x + y\), not just \(x\)! Don't stop after finding one variable.
โ๏ธ My Work:
3
INEQUALITIES
Algebra
โ
โ
โ
๐ง MEMORY KEY
FLIP the sign when multiplying/dividing by a NEGATIVE
Question: Which of the following is the solution set for \(-3x + 7 > 16\)?
A) \(x > -3\)
B) \(x < -3\)
C) \(x > 3\)
D) \(x < 3\)
โ ๏ธ
Most students forget to flip the inequality sign when dividing by \(-3\)!
โ๏ธ My Answer:
๐ SECTION 2 โ Quadratics & Polynomials
4
FACTORING QUADRATICS
Quadratic
โ
โ
โ
๐ง MEMORY KEY
Find 2 numbers: PRODUCT = c, SUM = b in \(x^2 + bx + c\)
๐ EXAMPLE
\(x^2 - 5x + 6 = 0\)
Find: product \(= 6\), sum \(= -5\) โ factors \(-2\) and \(-3\)
\((x-2)(x-3) = 0 \Rightarrow x = 2 \text{ or } x = 3\)
Find: product \(= 6\), sum \(= -5\) โ factors \(-2\) and \(-3\)
\((x-2)(x-3) = 0 \Rightarrow x = 2 \text{ or } x = 3\)
Question: What are the solutions to \(x^2 - 7x - 18 = 0\)?
A) \(x = 9\) and \(x = -2\)
B) \(x = -9\) and \(x = 2\)
C) \(x = 6\) and \(x = -3\)
D) \(x = 3\) and \(x = -6\)
โ๏ธ My Work:
5
QUADRATIC FORMULA
Quadratic
โ
โ
โ
๐ง MEMORY KEY
"minus b, plus or minus, root of b-squared minus 4ac, over 2a"
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Question: The equation \(2x^2 - 5x - 3 = 0\) has two solutions. What is their sum?
๐ก Sum of roots \(= -b/a\)
Product of roots \(= c/a\)
โ Vieta's shortcut!
Product of roots \(= c/a\)
โ Vieta's shortcut!
A) \(\dfrac{5}{2}\)
B) \(-\dfrac{3}{2}\)
C) \(\dfrac{3}{2}\)
D) \(-\dfrac{5}{2}\)
โ๏ธ My Work:
6
DISCRIMINANT
Quadratic
โ
โ
โ
๐ง MEMORY KEY
D = bยฒโ4ac โ (+) two roots ยท (0) one root ยท (โ) no real roots
Question: For what value of \(k\) does \(x^2 + kx + 9 = 0\) have exactly one real solution?
A) \(k = 3\)
B) \(k = 6\)
C) \(k = 9\)
D) \(k = 18\)
โ ๏ธ
One real solution means discriminant = 0, not > 0!
โ๏ธ My Work:
๐ SECTION 3 โ Functions
7
FUNCTION NOTATION
Functions
โ
โโ
๐ง MEMORY KEY
f(a) = "plug a into x" ยท f(f(x)) = "apply twice"
๐ EXAMPLE
If \(f(x) = 2x^2 - 1\), then \(f(3) = 2(9)-1 = 17\)
Question: If \(f(x) = 3x - 4\) and \(g(x) = x^2 + 2\), what is \(f(g(2))\)?
A) 14
B) 18
C) 20
D) 22
โ ๏ธ
Work inside out: find \(g(2)\) first, then apply \(f\). Never do \(g(f(2))\) by mistake!
โ๏ธ My Work:
8
LINEAR vs EXPONENTIAL
Functions
โ
โ
โ
๐ง MEMORY KEY
LINEAR: add same amount ยท EXPONENTIAL: multiply by same ratio
Question: A bacteria population doubles every 3 hours. If there are 500 bacteria initially, which expression gives the population after \(t\) hours?
A) \(500 + 2t\)
B) \(500 \cdot 2^t\)
C) \(500 \cdot 2^{t/3}\)
D) \(500 \cdot 3^{t/2}\)
โ ๏ธ
The doubling period is 3 hours, so the exponent is \(t/3\), not just \(t\)!
โ๏ธ My Answer:
๐ SECTION 4 โ Ratios, Rates & Percentages
9
PERCENT CHANGE
Percent
โ
โ
โ
๐ง MEMORY KEY
% Change = \(\dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100\)
Question: A shirt originally costs $80. It is first discounted by 20%, then the sale price is increased by 25%. What is the final price?
A) $80
B) $84
C) $76
D) $88
โ ๏ธ
โ20% then +25% โ +5%! Apply each percent to the updated price, not the original.
โ๏ธ My Work:
10
PROPORTIONS & UNIT RATES
Ratio
โ
โโ
๐ง MEMORY KEY
CROSS-MULTIPLY: \(\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc\)
Question: A car travels 150 miles in 2.5 hours. At the same speed, how many miles will it travel in 4 hours?
A) 220
B) 230
C) 240
D) 250
โ๏ธ My Work:
๐ SECTION 5 โ Geometry & Trigonometry
11
SIMILAR TRIANGLES
Geometry
โ
โ
โ
๐ง MEMORY KEY
SIMILAR = same shape, different size โ corresponding sides PROPORTIONAL
Question: Triangle ABC is similar to triangle DEF. If \(AB = 6\), \(BC = 9\), and \(DE = 10\), what is the length of \(EF\)?
A) 12
B) 15
C) 18
D) 20
โ ๏ธ
Match the corresponding sides correctly before setting up the proportion!
โ๏ธ My Work:
12
CIRCLE โ ARC & SECTOR
Geometry
โ
โ
โ
๐ง MEMORY KEY
\(\dfrac{\theta}{360} = \dfrac{\text{arc length}}{2\pi r} = \dfrac{\text{sector area}}{\pi r^2}\)
All three ratios are equal โ one formula connects them all.
Question: A circle has radius 12. An arc is subtended by a central angle of \(120ยฐ\). What is the length of the arc?
A) \(4\pi\)
B) \(6\pi\)
C) \(8\pi\)
D) \(10\pi\)
โ๏ธ My Work:
13
PYTHAGOREAN THEOREM
Geometry
โ
โโ
๐ง MEMORY KEY
\(a^2 + b^2 = c^2\) โ c is always the HYPOTENUSE (longest side)
Special triangles: 3-4-5 ยท 5-12-13 ยท 8-15-17
Question: In right triangle \(PQR\), the hypotenuse \(PR = 26\) and leg \(PQ = 10\). What is the length of leg \(QR\)?
A) 20
B) 24
C) 22
D) 16
๐ก Try multiplying 5-12-13 by 2 โ 10-24-26! Recognize the pattern and save time!
โ๏ธ My Work:
14
TRIGONOMETRY โ SOH CAH TOA
Trig
โ
โ
โ
๐ง MEMORY KEY
SOH: sin = Opp/Hyp ยท CAH: cos = Adj/Hyp ยท TOA: tan = Opp/Adj
Question: In right triangle \(ABC\) where \(\angle C = 90ยฐ\), \(AB = 13\) and \(BC = 5\). What is \(\sin(\angle A)\)?
A) \(\dfrac{5}{13}\)
B) \(\dfrac{12}{13}\)
C) \(\dfrac{5}{12}\)
D) \(\dfrac{13}{12}\)
โ ๏ธ
First find the missing side! Then be sure to identify which side is "opposite" to angle A vs angle B.
โ๏ธ My Work:
๐ SECTION 6 โ Statistics & Data Analysis
15
MEAN, MEDIAN, MODE
Statistics
โ
โโ
๐ง MEMORY KEY
MEAN = sum รท n ยท MEDIAN = middle value (sorted!) ยท MODE = most frequent
Question: The ages of 7 students are: 14, 16, 15, 17, 14, 18, 16. If a new student of age 20 joins the group, which measure of central tendency changes the MOST?
A) Mean only
B) Median only
C) Mode only
D) Mean and median equally
โ ๏ธ
Extreme values (outliers) affect the mean much more than the median or mode.
โ๏ธ My Work:
16
SCATTERPLOT & LINE OF BEST FIT
Data
โ
โ
โ
๐ง MEMORY KEY
y = mx + b: m = SLOPE (rate of change) ยท b = y-intercept (starting value)
Question: A line of best fit for a dataset is modeled by \(y = 2.4x + 5.1\), where \(x\) is hours studied and \(y\) is test score. According to this model, approximately how many points does a student's score increase for each additional hour of study?
A) 2.4 points
B) 5.1 points
C) 7.5 points
D) 12 points
โ ๏ธ
The slope (2.4) = rate of change. The y-intercept (5.1) is the starting value, not the rate!
โ๏ธ My Answer:
17
PROBABILITY
Probability
โ
โ
โ
๐ง MEMORY KEY
P(A) = favorable outcomes รท total outcomes ยท P(A and B) = P(A) ร P(B) if independent
Question: A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If two marbles are drawn one at a time without replacement, what is the probability that both marbles are red?
A) \(\dfrac{1}{11}\)
B) \(\dfrac{2}{11}\)
C) \(\dfrac{1}{9}\)
D) \(\dfrac{4}{33}\)
โ ๏ธ
"Without replacement" = after drawing 1 red, only 3 red remain out of 11 total. Don't use 4/12 twice!
โ๏ธ My Work:
๐ SECTION 7 โ Advanced Topics
18
EXPONENT RULES
Exponents
โ
โ
โ
๐ง MEMORY KEY
Multiply โ ADD exponents | Divide โ SUBTRACT | Power of power โ MULTIPLY
\(x^0 = 1\) | \(x^{-n} = \dfrac{1}{x^n}\) | \(x^{1/2} = \sqrt{x}\)
\(x^0 = 1\) | \(x^{-n} = \dfrac{1}{x^n}\) | \(x^{1/2} = \sqrt{x}\)
Question: Simplify: \(\dfrac{x^3 \cdot x^{-5}}{x^{-4}}\)
A) \(x^2\)
B) \(x^{-6}\)
C) \(x^{-2}\)
D) \(x^4\)
๐ STEP-BY-STEP
Numerator: \(x^3 \cdot x^{-5} = x^{3+(-5)} = x^{-2}\)
Then: \(x^{-2} \div x^{-4} = x^{-2-(-4)} = x^{\;?}\)
Then: \(x^{-2} \div x^{-4} = x^{-2-(-4)} = x^{\;?}\)
โ๏ธ My Work:
19
WORD PROBLEMS โ MODELING
Modeling
โ
โ
โ
๐ง MEMORY KEY
DEFINE variable โ WRITE equation โ SOLVE โ CHECK units!
Question: A company rents bikes for a $10 flat fee plus $3 per hour. A customer has a budget of at most $40. Which inequality represents the maximum number of hours \(h\) the customer can rent?
A) \(3h + 10 \leq 40\)
B) \(3h + 10 \geq 40\)
C) \(10h + 3 \leq 40\)
D) \(10h + 3 \geq 40\)
โ ๏ธ
"At most $40" โ โค 40. "At least" โ โฅ. Don't mix up the direction of the inequality!
โ๏ธ My Answer:
20
IMAGINARY NUMBERS
Complex
โ
โ
โ
๐ง MEMORY KEY
\(i = \sqrt{-1}\) ยท \(i^2 = -1\) ยท \(i^3 = -i\) ยท \(i^4 = 1\) โ then it cycles again!
๐ EXAMPLE
\(i^{13} = i^{12} \cdot i = (i^4)^3 \cdot i = 1^3 \cdot i = i\)
Divide the exponent by 4 โ use the remainder: 0โ1, 1โi, 2โโ1, 3โโi
Divide the exponent by 4 โ use the remainder: 0โ1, 1โi, 2โโ1, 3โโi
Question: What is the value of \((3 + 2i)(1 - 4i)\)?
A) \(3 + 10i\)
B) \(11 - 10i\)
C) \(-5 - 10i\)
D) \(11 + 10i\)
โ ๏ธ
Use FOIL! The key step: \(2i \cdot (-4i) = -8i^2 = -8(-1) = \)\(+8\). Don't forget \(i^2 = -1\)!
โ๏ธ My Work:
๐ QUICK REFERENCE
Algebra
โข Flip sign รท negative
โข Sum of roots = โb/a
โข Discriminant โ \(b^2โ4ac\)
โข Flip sign รท negative
โข Sum of roots = โb/a
โข Discriminant โ \(b^2โ4ac\)
Geometry
โข SOH CAH TOA
โข Arc: ฮธ/360 ร 2ฯr
โข Special triangles: 3-4-5
โข SOH CAH TOA
โข Arc: ฮธ/360 ร 2ฯr
โข Special triangles: 3-4-5
Functions
โข f(g(x)): inside out
โข Exponential: multiply ratio
โข Linear: add same amount
โข f(g(x)): inside out
โข Exponential: multiply ratio
โข Linear: add same amount
Stats / Data
โข Outlier affects mean most
โข Slope = rate of change
โข Without replacement: update!
โข Outlier affects mean most
โข Slope = rate of change
โข Without replacement: update!
โ๏ธ Score: ______ / 20 Date finished: ______________