Algebra 2 Semester 2
What is exponentiation and an antilog?
"when you are given the logarithm of the number and need to find the number." the number in exponentiation; eg. log x=b x=antilog b logb x = bˣ
How do you rewrite a quadratic in standard form in vertex form?
(A.) Divide away any coefficients from (y/x) 1. Move the c term to the other side 2. Complete the square: -a. do (b/2)², and ADD IT TO BOTH SIDES -b. create squared binomial 3. move estranged c term back to other side
What is radian measure of an angle?
(length of the arc on the unit circle subtented by the angle.)
The equation for a circle with center (h, k) and radius r is as follows:
(x-h)²+(y-k)²=r²
What is the equation of a circle with a center (h,k) and radius r?
(x-h)²+(y-k)²=r²
Parabolic Smileys
+ +. .- - . ∪ . . ∩
WHAT IS 0 FACTORIAL?
1
How do you find... 1. all possible rational zeros 2. all rational zeros 3. all irrational zeros
1. do the p/q thing 2. the successful p/q's put into synthetic division 3. try the unsuccessful ones(?)
What is the length of the major axis? What is the length of the minor axis? What is the distance between the foci?
2a 2b (a>b) 2c
What is the estimated measure of a radian?
57.2958... degrees
What is Pascal's Triangle
A triangle whom two of it's descending values add together... REMEMBER! 1's are 0th elements! use to do binomial expansions
How do you solve systems of conics?
A: Substitute one of the values, and factor into a quadratic. Then use the quadratic formula B: Eliminate and then solve for a linear solution ***
How do you simplify an equation of a circle?
A: With x²,x and y²,y terms: B: With ax² and ay² terms: 1. Divide the a term on both sides
How do you simplify ellipse equations?
A: ax²+bx+cy²+dy=e Complete the square... (WHEN ADDING C TERM TO BOTH SIDES, MULTIPLY IT BY THE COEFFICIENT FIRST)
Don't forget when completing the square...
ADD THE C TERMS TO THE OTHER SIDE TOO!
What is the equation of a conic section?
Ax²+Bxy+Cy²+Dx+Ey+F=0 where A,B, and C are not all 0
What is the Characteristic and Mantissa of a log?
Characteristic: number before the decimal Mantissa: numbers after the decimal WHEN IT IS LESS THAN 1, IT IS ±1 AND A DECIMAL THAT IS THE DIFFERENCE OF THE ORIGINAL DECIMAL
What is the equation of a combination
Combinations of n objects, taking r at a time, is denoted nCr and ⬞ ⬞ ⬞ ⬞ n! nCr=------ ⬞ ⬞ ⬞ r!(n-r)!
What is the distance formula?
D=√(x₁-x₂)²+(y₁-y₂)²
What are the proper forms of... Direct Variation Inverse Variation Joint Variation
Direct Variation: x₁ ⬞ ⬞ x₂ -- = -- y₁ ⬞ ⬞ y₂ Inverse Variation: x₁ ⬞ ⬞ x₂ -- = -- y₂ ⬞ ⬞ y₁ (improper if need be: x₁y₁=x₂y₂) Joint Variation: y₁ ⬞ ⬞ x₁z₁ -- = ---- (improper if need be: y=kxz) y₂ ⬞ ⬞ x₂z₂ (k is the variation constant)
What is normal distribution?
Distribution along a symmetrical bell curve
How do you use synthetic substitution to find f(a)
Divide by a. The answer is the remainder!
What is the equation of probability distribution
E(x)=∑x*p(x)
What are parts of a hyperbola
Focus (-ices) - (like in a parabola) Vertex (-ices) - (like in a parabola) Transverse axis - Axis that connects the vertices Conjugate axis- Axis perpendicular to the Transverse
What is the binomial theorem?
For all positive integers n... ⬞ n ⬞ ⬞ ⬞ ⬞ n! ⬞ ∑------------aⁿ⁻ᵏbᵏ=(a+b)ⁿ k=0 ⬞⬞ k!(n-k)!
What is the equality property of exponential functions?
For all positive numbers b where b≠1 and all real numbers a and c, bᵃ=bᶜ if and only if a=c
What is the equation of an ellipse with a center (h,k)... With a horizontal major axis? With a vertical major axis?
Horiz: (x-h)² ⬞ ⬞ (y-k)² ----- + ----- = 1 ⬞ a² ⬞ ⬞ ⬞ ⬞ b² Vert: (x-h)² ⬞ ⬞ (y-k)² ----- + ----- = 1 ⬞ b² ⬞ ⬞ ⬞ ⬞ a² where b²=a²-c² and a≠b
What is the equation of a hyperbola with a center (h,k)... With a horizontal transverse axis? With a vertical transverse axis?
Horizontal: (x-h)² ⬞ ⬞ (y-k)² ----- — ----- = 1 ⬞ a² ⬞ ⬞ ⬞ ⬞ b² Vertical: (y-k)² ⬞ ⬞ (x-h)² ----- — ----- = 1 ⬞ a² ⬞ ⬞ ⬞ ⬞ b² Where the distance between foci is c, c²=a²+b² (or xy=c when c≠0)
Permutation or combination? or factorial?
How many 𝒹𝒾𝒻𝒻𝑒𝓇𝑒𝓃𝓉 ways or how many ways use factorial for permutation when both n and r are the same value
What is the fundamental counting principle
If an event can occur in m different ways and another event can occur in n different ways, there are m * n ways both events can occur.
How do you remember if an ellipse is horizontal or vertical?
If the a term (BIGGEST) leads, then the ellipse is wide. If the b term (SMALLEST) leads, then the ellipse is thin.
What is the rational zero theorem?
If you have a polynomial function f(x) with integral coefficients in descending order, where p is a factor of the constant of f(x) and q is a factor of the leading coefficient of f(x), then p/q is a possible zero of f(x) (the last term/constant factors divided by the leading terms factors)
What is the mean, median and mode? And range?
Mean: Average. Median: Middlemost value Mode: Most common value Range: Difference of greatest quantity and lowest quantity
How do you use LCD to simplify rational expressions (no equals sign)?
Multiply each fraction by the others denominator ⬞⬞ a ⬞ ⬞ c ⬞ -- + -- ⬞x+b ⬞ x+d add
How do you find the vertices of a hyperbola?
Normal: (h-a,k) and (h+a,k) Sideways: (h,k-a) and (h,k+a)
Parts of a (right) triangle?
Opposite: opposite the reference angle Adjacent: adjacent to reference angle Hypotenuse: always the longest side
What is the equation of the probability of dependent events What is the equation of the probability of independent events?
P(A and B)=P(A)*P(B following A) P(a and B)=P(A)*P(B)
What are the discriminants for the conic sections of... Parabola Hyperbola Circle Ellipse
Parabola: B²-4AC=0 Hyperbola: B²-4AC>0 Circle: B²-4AC<0 , B=0 and A=C Ellipse: B²-4AC<0 and either B≠0 or A≠0
What is the equation of a permutation
Permutations of n objects, taking r at a time, is denoted nPr and ⬞ ⬞ ⬞ ⬞n! nPr=----- ⬞ ⬞ ⬞ (n-r)!
How do you use descartes rule of signs to determine... The number of... possible positive zeros negative zeros. Imaginary zeros.
Possible positive: eg. Ax^3-Bx^2+Cx-D Number of sign changes or number of sign changes decreased by an even number Possible negative: same but for f(-x) (*only the odd terms change) Possible imaginary: make a PNI chart: eg. Ax^3-Bx^2+Cx-D P.| 3 1 N| 0 0 I..| 0 2 IMAGINARY IS ALWAYS IN PAIRS DEGREE DETERMINES NUMBER OF ZEROS
What are some properties of logarithms?
Product of Logarithms property For any positive numbers b, u, and v and b≠1, logb uv = logb u + logb v Power of logarithms property For any positive numbers b, u, and v and b≠1, logb uᵛ=vlogb u ~others:~
What are some properties of exponents?
Product of powers: aᵐ*aⁿ=aᵐ⁺ⁿ Power of a Power property: (aᵐ)ⁿ=aᵐⁿ
What are some properties of quotients?
Quotient of powers property For any real number a (except 0), and integers m and n, aᵐ -- = aᵐ⁻ⁿ aⁿ Quotients of logarithms property For any positive numbers b, u, and v and b≠1, ⬞ ⬞ ⬞ u logb---=logb u-logb v ⬞ ⬞ ⬞ v
How to find: Random Variable Probability Distribution Expected Values
Random Variable: a rule that assigns a real number to each possible outcome in the sample space, denoted as X Probability Distribution Function:
How do you find the focus of a parabola?
Regular: (h , k+¼ₐ) Sideways: (h+¼ₐ , k)
What are the foci of hyperbolas with a center of (h,k)
Regular: (h-c,k) and (h+c,k) Sideways: (h,k-c) and (h,k+c)
How do you find the directrix of a parabola
Regular: y=k-¼ₐ Sideways: x=h-¼ₐ
What is the equation of the law of sines?
SinA ⬞ SinB ⬞ SinC ----=----=---- ⬞ a ⬞ ⬞⬞ b ⬞ ⬞⬞ c
What is the equation of an arithmetic mean
Sn -- n
How does Sigma notation work?
Sn = sum of all terms to nth term is a ☜is the last n ∑ Xn n=b ☜This is n₁
How do you find the foci of an ellipse?
Solve for c with c²=a²-b² then add, subtract from: ℍorizontal term in a ℍorizontal ellipse or 𝒦 term in a 𝒱ertical ellipse
How do you find factors of polynomials with... Long/Synthetic division? The remainder theorem? The factor theorem?
Step A: If no binomial is given, divide all possible factors of the constant term by all possible factors of the leading term. Try all of these in synthetic division. DONT FORGET TO USE THE COEFFICIENTS FOR ALL TERMS! eg. x³-7x-6 : a| 1 𝟎 -1 -6 ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞‾‾‾‾‾‾‾‾‾‾‾ Synthetic division: Divide (Long/Synthetically) by the factor/function value to get the result SOME TOOLS: Remainder theorem: f(x)=q(x)*(x-a)+f(a) or dividend = quotient * divisor + remainder Where q(x) is a polynomial whose degree is one less than the degree of f(x) Factor theorem: The binomial x-a is a factor of the polynomial f(x) if and only if f(a)=0
What are the parts of an ellipse? Short axis? Long axis? What/Where are the foci?
The Minor axis The Major axis Two special points, on the major axis
How do you approximate logs given two logs of the same base
Use law of logs
What is factorial?
When you get each descended quantity of a number and multiply them together. NOTE: 0!=1
How do you find an inverse function?
You switch x and y in the function, then solve for y.
What is the Direction of opening for quadratics in the form of y=a(x-h)²+k (regular) x=a(y-k)²+h (sideways)
[Regular:] Up if a>0 Down if a <0 [Sideways:] Right if a>0 Left if a <0
What is the equation of an arithmetic sequence?
an=a₁+(n-1)d Where a₁ is the first term, an is the nth term, and d is the common difference
What is the equation of a geometric series?
an=a₁rⁿ⁻¹ Where a₁ is the first term, an is the nth term, r is the common ratio, and r≠0
What is the relationship of a, b, & c in an ellipse.
b²=a²-c² (it involves the pythagorean theorem : )
What is the equation of the law of consines?
c²=a²+b²-2ab(cosC)
What is the proper form of an inverse function
f⁻¹(x)=
What does [g∘f ](x) mean
g(f(x))
Logarithm with base b
if b and y are positive numbers and b≠1, then log_b(y)=x if any only if bˣ = y.
Natural logs?
log to the base ℯ lnℯ = 1 ln = logℯ
Logarithms.
logₐx=y aʸ=x
How do you solve for an arithmetic sum in sigma notation?
num ∑(n something) n=1 do the thing in the parenthesis substituting n. add these together, each where n is the term number.
How do you convert between radians?
rad=degrees*π ⬞ ⬞ ⬞--------- ⬞ ⬞ ⬞ ⬞ ⬞180 degrees=rad*180 ⬞ ⬞ ⬞ ⬞ ⬞ ⬞------ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ π
What is the pythagorean identity?
sin²(θ)+cos²(θ)=1 (use it to find sin(θ), cos(θ), or tan(θ), given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle)
How do you use pascal's triangle for binomial expansion?
take the row for the degree of polynomial. take each term of the binomial and put it in parentheses. put the first term in a row under the values in the triangle row in descending powers (0-x) put the second term in below the first terms in ascending powers SUM OF EXPONENTS ARE TO THE BINOMIALS POWER eg:
What is the equation of probability of Inclusive events?
to find P(A or B) P(A or B) = P(A)+P(B)-P(A and B)
How do you solve exponential functions of e?
use logs/nat logs
What is the Axis of Symmetry for the Vertices (h,k) (h,k)
x=h y=k
What is a quadratic equation in vertex form? (Reg and sideways)
y=a(x-h)²+k (regular) x=a(y-k)²+h (sideways)
How do you find the length of Latus Rectum?
|¹/ₐ| Units
How do you find the area of a triangle with trigonometry?
½b*a*sinC
How do you find arc length?
θ*r where θ is in radians
What are the slopes of the asymptotes ?
ᵇ/ₐ and −ᵇ/ₐ ᵃ/ⓑ and −ᵃ/ⓑ
What is the midpoint formula?
⎧x₁+x₂ , y₁+y₂⎫ ⎩‾‾2‾‾ ⬞ ‾‾2‾‾⎭
What is the equation of standard deviation?
⬞ ⬞ /(x₁-x̅)²+(x₂-x̅)²+...+(xn-x̅)² σ=√---------------------- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞n
What is the equation of an infinite geometric series
⬞ ⬞ ⬞ a₁ S=------ ⬞ ⬞ 1 - r Where a₁ = the first term, r= the common ratio, and -1<r<1
What is the change of base formula?
⬞ ⬞ ⬞ ⬞ logb x logₐx=------ ⬞ ⬞ ⬞ ⬞ logb a Where b can be any real number. Note that we can use this formula to convert natural logarithms as well: ⬞ ⬞ ⬞ ⬞ ln x logₐx=---- ⬞ ⬞ ⬞ ⬞ ln a
What is the equation of odds... For an event? Against an event?
⬞ ⬞ ⬞ ⬞ number of favorable outcomes Odds=--------------------------- ⬞ ⬞ ⬞ ⬞ number of unfavorable events ⬞ ⬞ ⬞ ⬞number of unfavorable events Odds=--------------------------- ⬞ ⬞ ⬞ ⬞number of favorable events
what is the equation of the probability of an event?
⬞ ⬞ ⬞ ⬞ number of favorable outcomes P(e)=------------------------------ ⬞ ⬞ ⬞ ⬞ number of possible outcomes where 0≤P(e)≤1
What are the 6 trigonometric functions
⬞ ⬞ ⬞ ⬞ opp sinθ=----- ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ ⬞ ⬞ adj cosθ=---- ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ ⬞ ⬞ opp tanθ=----- ⬞ ⬞ ⬞ ⬞ adj ⬞ ⬞ ⬞ ⬞ hyp cscθ=----- ⬞ ⬞ ⬞ ⬞ opp ⬞ ⬞ ⬞ ⬞ hyp secθ=----- ⬞ ⬞ ⬞ ⬞ adj ⬞ ⬞ ⬞ ⬞ adj cotθ=----- ⬞ ⬞ ⬞ ⬞ opp
Trigonometric ratios? (Assuming side names are in relation to A)
⬞ ⬞ ⬞ ⬞ ⬞ opp ⬞ ⬞ ⬞ a Sine A=------ =--- ⬞ ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ ⬞ c sin ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ adj ⬞ ⬞ b Cosine A=----- =--- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ c cos ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ opp ⬞ ⬞ a Tangent A=---- = --- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ adj ⬞⬞ ⬞ b tan ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ c Cosecant A=----- = -- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ opp ⬞ ⬞ a csc ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ hyp ⬞ ⬞ ⬞ c Secant A=---- = ⬞ -- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ adj ⬞ ⬞ ⬞ b sec ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ adj ⬞ ⬞ b Cotangent A=--- = -- ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ opp ⬞⬞ a cot
What is the equation of a geometric series? (The Sum)
⬞ ⬞ ⬞a₁-a₁rⁿ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞a₁-anr Sn=-------- or Sn=------------ ⬞ ⬞ ⬞ ⬞1-r ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ 1-r
What is the equation of arithmetic series? (arithmetic sum)
⬞ ⬞ ⬞n(a₁+an) ⬞ ⬞ ⬞ ⬞ n[2a₁+(n-1)d] Sn=-------- or Sn=------------ ⬞ ⬞ ⬞ ⬞ ⬞2 ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ ⬞ 2 d= the common difference Sn= the sum of the first n terms
