Spring Final Notes & Equation SheetAlgebra 2 Pre-AP

Unit 7 Logs:

Exponential Function:

ex=a

Natural Logarithmic Function:

lnea=x

No x-intercept…………………….ExpRange of all real numbers……....LogY-axis is an asymptote…………..LogHas a y-intercept………………....LogX-axis is an asymptote…………..LogRange is [0,∞)..............................Exp

Exponential Function

Logarithmic Function

bk=N logbN=k

Domain (-∞, ∞) [0,∞)

Range [0,∞) (-∞, ∞)

Asymptote y=0 ( x-axis) x=0 (y-axis)

X-int NONE (1,0)

Y-int (0,1) NONE

Unit 8 Logs 2:

Compound Interest Formula A=P(1+r/n)nt

Continuously Compounded Interest Formula A=Pert

Decibel Voltage Gain Formula D=10logE₀ E₁

(E₀= the output voltage)(E₁= the input voltage)

Unit 14:

Pascal’s Triangle: *** starts with row 0(n) denotes the r element in row n of Pascal’s triangle.(r)

Synthetic Division: The second method used to divide polynomials is called synthetic (short) division. This method is only used when a polynomial is divided by a first degree binomial of the form x-k, where the coefficient of x is 1.

Pascal’s Triangle(n=row)

(r=element)

(n) = n! (r) r!(n-r)!

I’s i¹ = I i² = -1 i³ = -I i⁴ = 1

Unit 15:

Polynomial Functions- Rational Zeros Theorem: Let f be a polynomial with integral coefficients. The only possible rational zeros of f are p/q where p is a factor of the constant term of f(x) and q is a factor of the leading coefficient.

Rational Zeros Theorem factors of constant + pfactors of leading coefficient - q

Even Degree & Positive Leading Coefficient as x→ +∞; y→ +∞as x→ -∞; y→ +∞

Even Degree & Negative Leading Coefficient as x→ +∞; y→ -∞as x→ -∞; y→ -∞

Odd Degree & Positive Leading Coefficient as x→ +∞; y→ +∞as x→ -∞; y→ -∞

Odd Degree & Negative Leading Coefficient as x→ +∞; y→ -∞as x→ -∞; y→ +∞

Unit 16:

Rational FunctionsN(x) & D(x) are POLYNOMIALS

f(x)= N(x) D(x)

Function f(x)

Inverse of a function f ¹(x)⁻

RF Vertical Asymptotes the 0’s of the denominator

RF Horizontal Asymptote: (h.a.)…..n < m…..n = m

…..n > m

h.a. is y=0h.a. is y= leading coefficient leading coefficient

NO h.a. (slant asymptote)...long div. poly.

RF Intercepts:…..y-int…..x-int

TO FIND:let x=0, solve for y

(if any) are the 0’s of the numerator

RF Holes hole if a 0 in num. matches a 0 in denom.To Determine: simplify func. & plug in the 0

Unit 12:

UNIT 12/13:

Distance Formula d=√(x₂ - x₁)² + (y₂ - y₁)²

Midpoint Formula ( (x₁ + x₂), (y₁ + y₂) ) 2 2

Slope Formula m= y₂ - y₁ x₂ - x₁

Point-Slope Formula (lin. eq.) y - y₁ = m(x - x₁)

Slope-Intercept Formula (lin. eq.) y=mx+b

Name of Formula/Equation Formula/Equation

Perpendicular Lines slopes are - reciprocals of each other

Parallel Lines same slope

Line Tangent to a Circle touches circle at ONLY 1 point(perpendicular to the radius at that 1 point)

Perpendicular Bisector Equation must know midpoint of line and its slope

Standard Form of a Circlecenter of circle is (h,k); radius of circle is r

(x - h)² + (y - k)² = r²

Circle (radius r) @ Origin x² + y² = r²

Find Center of Circle & Radius?(in x²+y²+ax+by+c=0 form)

Completing The Square

Standard Form of a Circlecenter of circle is (h,k); radius of circle is r

(x - h)² + (y - k)² = r²

Circle (radius r) @ Origin x² + y² = r²

Find Center of Circle & Radius?(in x²+y²+ax+by+c=0 form)

Completing The Square

Center of Ellipse midpoint of F₁ and F₂

Ellipse Standard Form x² + y² = 1 a² b²

Vertices of Ellipse (major axis) (a,0) and (-a,0)

Co-Vertices of Ellipse (minor axis) (b,0) and (-b,0)

Foci of Ellipse (c,0) and (-c,0)

Ellipse Values (equation) c² = a² - b²

Equation for Conics: Ax2 + Bxy + Cy2 + Dx + Ey + F =0

An ellipse if… B2 – 4 AC < 0A parabola if… B2 – 4 AC = 0A hyperbola if… B2 – 4 AC > 0

Unit 17:

Sequence = (,) or (.)

Series = (+)

Arithmetic

Recursive: a1 = ? an = an-1 + d

Explicit: an = a1 + (n-1)d

Sum of the Series: Sn = n/2(a1 + an) OR Sn = n/2(2a1 + (n-1)d)

Geometric

Recursive: g1 = ? gn = (gn-1)r

Explicit: gn = g1(rn-1)

Sum of a Finite/Infinite Series: Sn = g1 (1-r n ) OR Sn = g1 1-r 1-r

Sigma Notation: 7 # of terms

Σ 3n + 8 arithmetic explicit n = 1 start term

Unit 18-Trig. Identities: