Number & Quantity
10%of exam
Real NumbersComplex NumbersRational ExponentsUnits & RatesRatios
Algebra
20%of exam
EquationsInequalitiesSystemsFactoringQuadraticsPolynomials
Functions
20%of exam
Domain & RangeTransformationsExponentialLogarithmicTrigonometricComposition
Calculus
10%of exam
Geometry
20%of exam
CongruenceSimilarityCirclesTransformationsCoordinateVolume
Probability & Statistics
20%of exam
Center & SpreadNormal DistributionConditional ProbabilityPermutations & CombinationsSampling
Quick Facts
- Test
- 5165
- Questions
- 66 selected-response
- Time
- 180 min
- Retake
- 28 days
- Pass
- Varies by state
- Fee
- $130
- Calculator
- On-screen graphing
- Blueprint
- 30 30 20 20
Discrete vs Continuous
Discrete
- Countable values
- Whole numbers
- Histograms
Continuous
- Interval range
- Real numbers
- Smooth curves
Count vs measure
Number System
- Real numbers
- Closure, commutativity, distributivity
- i^2
- Equals -1Complex
- a^(m/n)
- Root then power
- |A| = k
- Split A=k and A=-k
- Principal root
- Non-negative root
- Dimensional analysis
- Cancel units multiply by 1
- Ratio vs rate
- Same vs different units
Order of Operations
Parentheses Exp Mult Div Add Sub
Left to rightExp before multMult before add
Algebra Equations
- Linear equation
- Solve isolate variable
- Inequality
- Flip on negative multiply
- System
- Elimination or substitution
- a^2 - b^2
- (a-b)(a+b)Factor
- Quadratic formula
- x = -b +/- sqrt / 2a
- Discriminant
- b^2 - 4ac sign
- Completing square
- Vertex form a(x-h)^2+k
- Factor theorem
- p(r)=0 implies (x-r)
Quadratic Discriminant
Positive two, zero one, negative none
D>0: two realD=0: repeatedD<0: complex
Right Triangle Trig
SOH CAH TOA
sin=opp/hypcos=adj/hyptan=opp/adj
Domain vs Range
Domain
- Valid inputs
- Avoid zero denom
- No neg even root
Range
- Output values
- Depends on family
- Check extrema
Input vs output
Function Family Picker
- Constant rate→Linear(y = mx + b)
- Curved growth→Quadratic(Parabola)
- Percent change→Exponential(y = ab^x)
- Inverse proportion→Rational(y = k/x)
- Repeating pattern→Trigonometric(Sine/cosine)
- Log of product→Logarithmic(Inverse exp)
Functions Core
- Function
- One input one output
- Vertical line test
- Detects function
- Domain
- Valid inputs
- Range
- Resulting outputs
- f(g(x))
- Apply g then f
- Inverse f^-1
- Reverses mapping
- Even function
- f(-x) = f(x)
- Odd function
- f(-x) = -f(x)
Graphing & Families
- f(x)+k
- Shift up k
- f(x-h)
- Shift right h
- y = ab^x
- Exponential growth/decay
- log_b(y) = x
- b^x = y
- Average rate
- Slope of secant
- Asymptote
- Line not crossed
Trigonometry
- Unit circle
- cos=x, sin=y
- SOH CAH TOA
- sin=opp/hyp etc
- Pythagorean identity
- sin^2 + cos^2 = 1
- Radian
- 2pi = 360 degrees
- Reference angle
- Acute to x-axis
- Reciprocal id
- csc = 1/sin
Fundamental Theorem
Derivative undoes integral
d/dx integral = fIntegral = F(b)-F(a)
Derivative vs Integral
Derivative
- Rate of change
- Slope tangent
- Power rule
Integral
- Accumulated area
- Antiderivative
- FTC link
Rate vs total
Derivative Rule Picker
- x^n→Power rule(n x^(n-1))
- Product f*g→Product rule(fg'+f'g)
- Quotient f/g→Quotient rule((f'g-fg')/g^2)
- Nested f(g(x))→Chain rule(f'(g)*g')
- e^x→Itself(Derivative = e^x)
- sin(x)→cos(x)(Chain applies)
Derivatives
- f'(x)
- Limit of difference quotient
- Power rule
- d/dx x^n = n x^(n-1)
- Product rule
- fg' + f'g
- Quotient rule
- (f'g - fg')/g^2
- Chain rule
- f'(g(x)) * g'(x)
- Critical point
- f'=0 or undefined
- Second derivative
- Concavity test
Integrals
- Antiderivative
- Reverse derivative
- Definite integral
- Signed area under curve
- FTC part 1
- d/dx integral = f(x)
- FTC part 2
- Integral = F(b)-F(a)
- u-substitution
- Reverse chain rule
Congruent vs Similar
Congruent
- Same size shape
- Rigid transforms
- SSS SAS ASA
Similar
- Same shape only
- Allows dilation
- AA criterion
Equal vs proportional
Euclidean Geometry
- SSS/SAS/ASA
- Congruence criteriaCongruence
- AA
- Similarity criterion
- a^2+b^2=c^2
- Right trianglePythagoras
- Transversal
- Corresponding angles equal
- Inscribed angle
- Half central angle
- C = 2*pi*r
- Circumference
- A = pi*r^2
- Circle area
- V = 4/3 pi r^3
- Sphere volume
Coordinate Geometry
- Distance formula
- sqrt (dx)^2 + (dy)^2
- Midpoint
- Average of coordinates
- y = mx + b
- Slope-intercept
- y-y1 = m(x-x1)
- Point-slope
- Rigid transform
- Preserves distances
- Dilation
- Scales distances
- V_cyl = pi r^2 h
- Cylinder volume
Normal Empirical Rule
68 95 99.7 within 1 2 3 sigma
1 sigma 68%2 sigma 95%3 sigma 99.7%
Permutation vs Combination
Permutation
- Order matters
- nPr
- Rankings, codes
Combination
- Order ignored
- nCr
- Committees, subsets
Sequence vs set
Center Measure Picker
- Symmetric data→Mean(Outlier sensitive)
- Skewed data→Median(Outlier resistant)
- Categorical data→Mode(Most frequent)
- Bell shaped→Mean = median(Normal distribution)
- Spread measure→Std deviation(Sqrt variance)
Statistics & Probability
- Mean
- Arithmetic average
- Median
- Middle value
- Mode
- Most frequent
- Std deviation
- Sqrt of variance
- Normal dist
- Bell, symmetric
- Empirical rule
- 68-95-99.7%
- P(A|B)
- P(A and B) / P(B)
- P(A and B)
- Independent: P(A)*P(B)
- E(X)
- Sum x*p(x)
- nPr
- Order matters
- nCr
- Order doesn't matter
Mean vs Median
Mean
- Arithmetic average
- Outlier sensitive
- Uses all values
Median
- Middle value
- Outlier resistant
- Position based
Average vs position
Common Traps
SSA/AAA excluded
SSA not congruence ≠ AAA only similarity
Correlation vs causation
Correlation is association ≠ Causation needs experiment
Inequality flip
Multiply negative reverses ≠ Add subtract preserves
Horizontal shift
f(x-h) shifts right ≠ h positive moves right
Root-then-power
Compute root first ≠ Then raise to power
Independent vs exclusive
Independent P(A|B)=P(A) ≠ Mutually exclusive P=0
Last Minute
- 1.Weights: 30/30/20/20
- 2.66 Q in 180 min
- 3.Retake wait 28 days
- 4.On-screen graphing calculator
- 5.Task of teaching ~25%
- 6.SOH CAH TOA
- 7.Discriminant: D>0 two real
- 8.FTC links derivative integral
- 9.nPr order; nCr no order
- 10.Mean sensitive; median resistant
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