Algebra review
Ch. 1
Point-Slope Form:
(y - y1) = m(x - x1)
Quadratic Formula:
x = (-b ± √(b2 - 4ac)) / (2a)
Difference Quotient:
(f(x + h) - f(x)) / h
Vertical Line Test → f(x) is not a function if the vertical line intercepts with more than one point
Perpendicular lines have negative reciprocals
Transformations | |
Linear | y = a(x -h) + k |
Quadratic | y = a(x - h)² + k |
Cubic | y = a(x - h)³ + k |
Square Root | y = a√(x - h) + k y = a(x - h)1/2 + k |
Reciprocal | y = a(1 / (x - h)) + k |
Exponential | y= ab(x - h) + k |
**The domain of a composite function is restricted by the domain of the input function
Inverse Functions
The composition of inverse functions is equal to 0 → f(f-1(x))=x
To find inverse functions, replace f(x), or y, with the x-variable and x with a y- . Then, solve for y, or f-1(x).
Ch 2 - Polynomials & Rational Functions
Graph of polynomial functions are continuous (no breaks, holes, or gaps)
Extrema: the minimums and maximums of a function
Relative/Local Extrema
Absolute/Global Extrema
∅ = Empty set
Between two real zeros, there must be at least one local min or max
Even-degree polynomials have either a global min or global max
Point of Inflection: occurs where the function changes from concave up to concave down, or vice versa
ROC changes from increasing to decreasing
Limits
End Behavior: what happens to the values of f(x) as x increases or decreases without bounds
To find, use leading coefficient test (e.g. LC is odd/even, negative/positive)
Zeros of a Polynomial
x = a → zero/solution of polynomial
(x - a ) → factor of polynomial
(a, 0) → x-intercept of polynomial
Imaginary Numbers
i = √(-1)
Complex Number: a + bi
Complex Conjugate: a - bi
To rationalize imaginary numbers (in denominators), multiply by the conjugate
Division Algorithm Theorem: f(x) = d(x)*g(x) + r(x)
where f(x) is = dividend, d(x) = divisor, g(x) = quotient, and r(x) = remainder
Remainder Theorem: If polynomial f(x) is divided by x-h, then the remainder is r = f(k)
Factor Theorem: A polynomial f(x) has a factor (x-h) if and only if f(k) = 0
Rational Root Test: If a polynomial f(x) has integer coefficients, then every rational zero of f(x) has the form p / q
where p = factor of constant term and q = factor of leading coefficient
Rational Functions
f(x) = N(x) / D(x) = anxn / bmxm
VA(s) are zeros of the denominator, HA is determined by the degrees of N(x) and D(x)
n < m → y = 0
n = m → y = LC/LC
n > m → no HA (slant asymptote, if n < m by 1)
SA is found by dividing N(x) by D(x)
Graphing Rational Functions
Simplify f(x), if possible, and list all restrictions
Find and plot y-intercepts
Find and plot the zeros of the numerator
Find the VA (zeros of denominator)
Find and sketch other asymptotes with a dashed line
Plot min. 2 points between asymptotes and one point beyond each x-intercept and VA
Use smooth curves to complete graph
Ch. 3 - Exponentials and Logs
Log Form: y = loga x
a must be >1 and positive
x must be positive
Exponential Form: ay = x
Prop. of Logs | |
Product Property |
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Quotient Property |
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Power Property |
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Change of Base Formula:
loga x = ln x / ln a
loga x = logb x / logb a
Logistic Model
Calculating Interest
Normal Interest: A = P(1 + (r / n))nt
Compound Interest: A = Pert
Ch. 4 - Trig Functions (radians)
Counter-clockwise (CCW) angle → +θ
Clockwise (CW) angle → -θ
Coterminal Angles: angles which share the same terminal side in standard positon
There’s an infinite number of coterminal angles
Arc Length Formula: s = θr
Unit Conversion: π rad = 180°
Trigonometric Functions Inverse Trig Functions | ||
sin θ = y / r | csc θ = r / y | arcsin (y / r) = θ |
cos θ = x / r | sec θ = r / x | arccos (x / r) = θ |
tan θ = y / x | cot θ = x / y | arctan (y / x) = θ |
**Solve the right triangle → find all angles and sides
**Remember to add 2π, n∈ℤ if there’s no restricted range
The Unit Circle
Trig Identities/Formulas
Half-Angle Quadrants
Parent Graphs
Sine Formula: y = a(sin(bx - c)) + d
Cosine Formula: y = a(cos(bx - c)) + d
Amplitude: (a) half of the distance between min/max values
Period (T) = 2π / b
Phase Shift = c / b
**MISTAKE: The midpoint of one tangent cycle is at the origin
Inverse Trig Function | Domain | Range |
y = arcsin x | [-1, 1] | [-π/2, π/2] → Q1 & Q4 |
y = arccos x | [-1, 1] | [0, π] → Q1 & Q2 |
y = arctan x | (-∞, ∞) | (-π/2, π/2) → Q1 & Q4 |
Law of Sines
Law of Cosines
Ch. 9 - Parametric & Polar Functions
Eliminating the Parametric
In one parametric equation, solve for t
Substitute the equation for t in the other equation
Simplify
Polar Coordinates
Coordinate Conversion
sin θ = y / x
cos θ = x / r
tan θ = y / x
Polar-to-Rectangular
x = r(cos θ)
y = t(sin θ)
Rectangular-to-Polar
tan θ = y / x
x2 + y2 = r2
Testing for Symmetry in Polar Equations
Over line θ = pi/2: replace (r, θ) by (r, θ-π) or (-r, -θ)
The polar axis: replace (r, θ) by (r, -θ) or (-r, π-θ)
The pole: replace (r, θ) by (r, π+θ) or (-r, θ)