PreCalculus Formulas - MathBits.com

PreCalculus Formulas Sequences and Series: Binomial Theorem

⎛ n⎞ ( a + b) = ∑ ⎜ ⎟ a n− k b k k =0 ⎝ k ⎠ n

n

Find the rth term

⎛ n ⎞ n−( r −1) r −1 b ⎜ r − 1⎟ a ⎝ ⎠ Functions: To find the inverse function: 1. Set function = y 2. Interchange the variables 3. Solve for y

Arithmetic Last Term

Geometric Last Term

an = a1 + (n − 1)d

an = a1r

Geometric Partial Sum

Arithmetic Partial Sum

⎛1− rn ⎞ S n = a1 ⎜ ⎟ ⎝ 1− r ⎠

⎛a +a ⎞ Sn = n ⎜ 1 n ⎟ ⎝ 2 ⎠ f -1 (x)

n −1

Composition of functions:

( f g )( x) = f ( g ( x)) ( g f )( x) = g ( f ( x)) (f

[r (cosθ + i sin θ )]n = r n (cos niθ + i sin niθ )

r = a 2 + b2 b θ = arctan a

x = r cosθ y = r sin θ

a + bi i = −1 i 2 = −1

( r ,θ ) → ( x , y )

Determinants:

3 5 4 3

= 3i3 − 5i4

Use your calculator for 3x3 determinants.

−1

f )( x) = x

Algebra of functions: ( f + g )( x) = f ( x) + g ( x) ; ( f − g )( x) = f ( x) − g ( x)

( f i g )( x) = f ( x)i g ( x) ; ( f / g )( x) = f ( x) / g ( x), g ( x) ≠ 0 Domains:: D( f ( x)) ∩ D( g ( x)) Domain (usable x’s) Asymptotes: (vertical) Watch for problems with Check to see if the zero denominators and with denominator could ever be negatives under radicals. zero. x f ( x) = 2 Range (y’s used) x + x−6 Difference Quotient Vertical asymptotes at f ( x + h) − f ( x ) x = -3 and x = 2 h terms not containing a mult. of h will be eliminated.

Complex and Polars: DeMoivre’s Theorem:

Asymptotes: (horizontal) x+3 1. f ( x) = 2 x −2 top power < bottom power means y = 0 (z-axis) 4 x2 − 5 2. f ( x) = 2 3x + 4 x + 6 top power = bottom power means y = 4/3 (coefficients) x3 3. f ( x) = None! x+4 top power > bottom power

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Cramer’s Rule:

ax + by = c dx + ey = f

⎛c ⎜ a b⎝ f d e 1

b a , e d

c⎞ ⎟ f ⎠

Also apply Cramer’s rule to 3 equations with 3 unknowns.

Trig:

Reference Triangles:

o a o sin θ = ; cos θ = ; tan θ = h h a h h a csc θ = ; secθ = ; cotθ = o a o

BowTie

Analytic Geometry: Circle

Ellipse

( x − h) 2 + ( y − k ) 2 = r 2 Remember “completing the square” process for all conics.

Parabola

( x − h) = 4a ( y − k ) ( y − k ) 2 = 4a ( x − h) 2

Polynomials: Remainder Theorem: Substitute into the expression to find the remainder. [(x + 3) substitutes -3]

Descartes’ Rule of Signs 1. Maximum possible # of positive roots → number of sign changes in f (x) 2. Maximum possible # of negative roots → number of sign changes in f (-x)

vertex to focus = a, length to directrix = a, latus rectum length from focus = 2a

( x − h) 2 ( y − k ) 2 + =1 a2 b2 larger denominator → major axis and smaller denominator → minor axis

Hyperbola

( x − h) ( y − k) − =1 2 a b2 2

2

Latus length from focus b2/a

c → focus length where major length is hypotenuse of right triangle. Latus rectum lengths from focus are b2/a a→transverse axis b→conjugate axis c→focus where c is the hypotenuse. asymptotes needed

Synthetic Division Mantra: “Bring down, multiply and add, multiply and add…”

Depress equation

[when dividing by (x - 5), use +5 for synthetic division]

(also use calculator to examine roots)

Analysis of Roots P N C Chart

Upper bounds: All values in chart are + Lower bounds: Values alternate signs No remainder: Root

* all rows add to the degree * complex roots come in conjugate pairs * product of roots - sign of constant (same if degree even, opposite if degree odd) * decrease P or N entries by 2

−b ± b 2 − 4ac x= 2a

Sum of roots is the coefficient of second term with sign changed.

Eccentricity: e = 0 circle 0 < e < 1 ellipse e = 1 parabola e > 1 hyperbola

Induction:

Find P(1): Assume P(k) is true: Show P(k+1) is true:

Rate of Growth/Decay: y = y0 ekt y = end result, y0 = start amount, Be sure to find the value of k first.

Far-left/Far-right Behavior of a Polynomial The leading term (anxn ) of the polynomial determines the far-left/far-right behavior of the graph according to the following chart. (“Parity” of n Æ whether n is odd or even.) LEFT-HAND BEHAVIOR n is even n is odd

anxn

(same as right)

an > 0 RIGHTHAND always positive BEHAVIOR or

(opposite right)

negative x < 0 positive x > 0

Leading Coefficient Test

Product of roots is the constant term (sign changed if odd degree, unchanged if even degree).

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an < 0

always negative

positive x < 0 negative x > 0