Pre-Calculus Final Exam

Domain of rational function f(x) = g(x)/h(x)

All real numbers except where h(x) = 0. Set denominator equal to zero and solve for excluded values.

Domain of square root function f(x) = √g(x)

All real numbers where g(x) ≥ 0. Set expression under radical ≥ 0 and solve.

Domain of logarithmic function f(x) = log(g(x))

All real numbers where g(x) > 0. Set argument greater than zero and solve.

Range of quadratic f(x) = a(x-h)² + k where a > 0

[k, ∞) because parabola opens upward with vertex at (h,k)

Range of quadratic f(x) = a(x-h)² + k where a < 0

(-∞, k] because parabola opens downward with vertex at (h,k)

Horizontal transformation f(x-h)

Shift RIGHT h units if h > 0, shift LEFT |h| units if h < 0

Vertical transformation f(x) + k

Shift UP k units if k > 0, shift DOWN |k| units if k < 0

Reflection transformation -f(x)

Reflect over x-axis (flip upside down)

Reflection transformation f(-x)

Reflect over y-axis (flip left-right)

Vertical stretch/compression af(x) where a > 0

Stretch by factor a if a > 1, compress by factor a if 0 < a < 1

Horizontal stretch/compression f(bx) where b > 0

Compress by factor 1/b if b > 1, stretch by factor 1/b if 0 < b < 1

Vertical asymptotes in rational functions

Occur where denominator = 0 but numerator ≠ 0. Set denominator equal to zero and solve.

Horizontal asymptote when degree of numerator < degree of denominator

y = 0

Horizontal asymptote when degree of numerator = degree of denominator

y = (leading coefficient of numerator)/(leading coefficient of denominator)

Horizontal asymptote when degree of numerator > degree of denominator

No horizontal asymptote exists

Slant asymptote condition

Exists when degree of numerator = degree of denominator + 1. Find by polynomial long division.

Hole in rational function

Occurs when both numerator and denominator equal zero at the same x-value. Factor and cancel common factors.

Even function definition

f(-x) = f(x) for all x in domain. Graph is symmetric about y-axis.

Odd function definition

f(-x) = -f(x) for all x in domain. Graph is symmetric about origin.

Function composition (f∘g)(x)

f(g(x)). Substitute g(x) into every x in f(x).

Inverse function f⁻¹(x) definition

Function that undoes f(x). If f(a) = b, then f⁻¹(b) = a.

One-to-one function test

Passes horizontal line test. Each y-value corresponds to exactly one x-value.

Vertex form of parabola

f(x) = a(x-h)² + k where vertex is at (h,k)

Standard form to vertex form

Complete the square: ax² + bx + c = a(x + b/2a)² + (c - b²/4a)

Quadratic formula

x = (-b ± √(b²-4ac))/(2a) for ax² + bx + c = 0

End behavior of polynomial with positive leading coefficient and even degree

As x → -∞, f(x) → +∞; As x → +∞, f(x) → +∞

End behavior of polynomial with positive leading coefficient and odd degree

As x → -∞, f(x) → -∞; As x → +∞, f(x) → +∞

End behavior of polynomial with negative leading coefficient and even degree

As x → -∞, f(x) → -∞; As x → +∞, f(x) → -∞

End behavior of polynomial with negative leading coefficient and odd degree

As x → -∞, f(x) → +∞; As x → +∞, f(x) → -∞

Rational Root Theorem

Possible rational zeros of polynomial are ±(factors of constant term)/(factors of leading coefficient)

Complex number addition (a + bi) + (c + di)

(a + c) + (b + d)i

Complex number multiplication (a + bi)(c + di)

(ac - bd) + (ad + bc)i

Complex conjugate of a + bi

a - bi

Exponential function f(x) = ab^x properties

Domain: (-∞,∞), Range: (0,∞) if a > 0, horizontal asymptote y = 0

Logarithmic function f(x) = log_b(x) properties

Domain: (0,∞), Range: (-∞,∞), vertical asymptote x = 0

Logarithm properties: Product rule

logb(xy) = logb(x) + log_b(y)

Logarithm properties: Quotient rule

logb(x/y) = logb(x) - log_b(y)

Logarithm properties: Power rule

logb(x^n) = n·logb(x)

Change of base formula

log_b(x) = log(x)/log(b) = ln(x)/ln(b)

Compound interest formula

A = P(1 + r/n)^(nt) where P=principal, r=rate, n=compounds per year, t=time

Continuous compound interest formula

A = Pe^(rt)

Converting degrees to radians

Multiply by π/180. Example: 90° = 90·π/180 = π/2 radians

Converting radians to degrees

Multiply by 180/π. Example: π/3 = π/3·180/π = 60°

Coterminal angles

Angles that differ by multiples of 360° (or 2π radians)

Reference angle

Acute angle between terminal side and x-axis

Unit circle: sin(30°) or sin(π/6)

1/2

Unit circle: cos(30°) or cos(π/6)

√3/2

Unit circle: tan(30°) or tan(π/6)

√3/3 or 1/√3

Unit circle: sin(45°) or sin(π/4)

√2/2

Unit circle: cos(45°) or cos(π/4)

√2/2

Unit circle: tan(45°) or tan(π/4)

1

Unit circle: sin(60°) or sin(π/3)

√3/2

Unit circle: cos(60°) or cos(π/3)

1/2

Unit circle: tan(60°) or tan(π/3)

√3

Sine function properties

Domain: (-∞,∞), Range: [-1,1], Period: 2π, Amplitude: 1

Cosine function properties

Domain: (-∞,∞), Range: [-1,1], Period: 2π, Amplitude: 1

Tangent function properties

Domain: all real except odd multiples of π/2, Range: (-∞,∞), Period: π

Amplitude of f(x) = A sin(Bx + C) + D

|A|

Period of f(x) = A sin(Bx + C) + D

2π/|B|

Phase shift of f(x) = A sin(Bx + C) + D

-C/B units

Vertical shift of f(x) = A sin(Bx + C) + D

D units

Pythagorean identity

sin²θ + cos²θ = 1

Tangent identity

tan θ = sin θ/cos θ

Reciprocal identities

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Sine addition formula

sin(A + B) = sin A cos B + cos A sin B

Cosine addition formula

cos(A + B) = cos A cos B - sin A sin B

Sine subtraction formula

sin(A - B) = sin A cos B - cos A sin B

Cosine subtraction formula

cos(A - B) = cos A cos B + sin A sin B

Double angle formula: sin(2θ)

2 sin θ cos θ

Double angle formula: cos(2θ)

cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

Law of Sines

a/sin A = b/sin B = c/sin C

Law of Cosines

c² = a² + b² - 2ab cos C

When to use Law of Sines

AAS (Angle-Angle-Side), ASA (Angle-Side-Angle), SSA (Side-Side-Angle)

When to use Law of Cosines

SAS (Side-Angle-Side), SSS (Side-Side-Side)

Triangle area formula with two sides and included angle

Area = (1/2)ab sin C

Heron's formula for triangle area

Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2

Inverse sine function domain and range

Domain: [-1,1], Range: [-π/2, π/2]

Inverse cosine function domain and range

Domain: [-1,1], Range: [0, π]

Inverse tangent function domain and range

Domain: (-∞,∞), Range: (-π/2, π/2)

Signs of trig functions in Quadrant I

All positive (sin, cos, tan all +)

Signs of trig functions in Quadrant II

sin positive, cos negative, tan negative

Signs of trig functions in Quadrant III

sin negative, cos negative, tan positive

Signs of trig functions in Quadrant IV

sin negative, cos positive, tan negative

Arc length formula

s = rθ where s = arc length, r = radius, θ = central angle in radians

Sector area formula

A = (1/2)r²θ where θ is in radians

Function notation f(g(x))

Composition of functions: substitute g(x) into f

Difference quotient

[f(x+h) - f(x)]/h

Piecewise function evaluation

Use the piece of the function that corresponds to the given x-value's domain

Finding x-intercepts

Set f(x) = 0 and solve for x

Finding y-intercepts

Evaluate f(0)

Vertical line test

A graph represents a function if every vertical line intersects it at most once

Horizontal line test

A function is one-to-one if every horizontal line intersects its graph at most once

Maximum/minimum of quadratic f(x) = ax² + bx + c

Occurs at x = -b/(2a), value is f(-b/(2a))

Synthetic division setup

Use when dividing polynomial by (x - c). Put c in the box, coefficients in a row.