2nd semester Flashcards
Vertical equation of a parabola
(x-h)^2 = 4p(y-k)
Vertex of a parabola
(h,k)
Horizontal equation of a parabola
(y-k)^2 =4p(x-h)
Focus of a vertical parabola
(h, k+p)
Horizontal parabola focus
(h+p, k)
Vertical parabola directrix
y= k-p
Horizontal parabola Directrix
x=h-p
Latus rectum of a parabola
4p
Vertical parabola axis
x=h
Horizontal parabola axis
y=k
Horizontal ellipse equation
((x-h)^2/a^2) + ((y-k)^2/b^2) = 1
a >b
c^2 = a^2 - b^2
E= c/a
Vertical ellipse equation
((x-h)^2/b^2) + ((y-k)^2/a^2) = 1
a >b
c^2 = a^2 - b^2
E= c/b
Center of an ellipse
(h, k)
Horizontal ellipse foci
(h+-c, k)
Vertical ellipse foci
(h, k+-c)
Horizontal ellipse vertices
(h+-a, k)
Vertical ellipse vertices
(h, k+-a)
Horizontal ellipse covertices
(h, k+-b)
Vertical ellipse covertices
(h+-b, k)
Horizontal ellipse major axis
2a
Horizontal ellipse minor axis
2b
Vertical ellipse major axis
2b
Vertical ellipse minor axis
2a
Horizontal hyperbola equation
((x-h)^2/a^2) - ((y-k)^2/b^2) = 1
c^2 = a^2 + b^2
E = c/a
Vertical hyperbola equation
((y-k)^2/a^2) - ((x-h)^2/b^2) = 1
c^2 = a^2 + b^2
E = c/a
Hyperbola center
(h, k)
Horizontal hyperbola Vertices
(h+-a, k)
Vertical hyperbola vertices
(h, k+-a)
Horizontal hyperbola foci
(h+-c, k)
Vertical hyperbola foci
(h, k+-c)
Horizontal hyperbola slope
+- b/a
Vertical hyperbola slopes
+- a/b
A function is even if…
The ends face the same way
Regardless of the degree or number of zeros
A function is odd if…
It’s ends face the opposite way
Regardless of the degree or number of zeros
Decartes rule of signs
If a+bi is a zero of the function, then a-bi is also a zero
Rational zero theorem
p/q = factors of constant/ factors of leading coefficient
p/q are possible zeros
Vertical asymptotes
What makes the denominator of a function zero, and does not have removable discontinuity (hole)
Horizontal asymptotes
Ax^n / Bx^m
n < m : x-axis is asymptote
n = m : fraction of leading coefficients is asymptote
n > m : no asymptote
Direct variation
Where k= y/x
x1/y1 = x1/y1
Inverse variation
where k= xy
x1/y2 = x2/y1
Joint variation
Where y/xz = k
y1/y2 = x1z1/x2z2
When solving rational functions and absolute value functions…
Always check your answer because it may not work out when you plug it back in
Exponential functions
Y= a(1+r)^t a= initial amount r= percent increase (expressed as a decimal) t= time
Logarithmic function
Log(b) y=x
b^x = y
Product of powers property
a^m x a^n = a^(m+n)
Power of a power property
(a^m)^n = a^mn
Product of logarithms property
Log(b) uv = log(b) u + log(b) v
Power of logarithms property
Log(b) u^v = vlog(b) u
Quotient of powers property
a^m/ a^n = a^(m-n)
Quotients of logarithms property
Log(b) u/v = log(b) u - log(b) v
Characteristics
Numbers in front of the decimal
Mantissa
Numbers behind the decimal
If it is negative add ten then subtract ten
Antilog
10^x
Interest formula
A = Pe^rt A: total amount P: principal r: interest rate t: time in years e: natural base
Antilogrithms
e^b = ? Or ln? = b
Logarithms
Ln (n) = ? or e^? = n
Compounded interest formula
A = P(1 + r/n)^nt A: total amount P: principal amount r: interest rate ( in decimal form) t: time in years n: number of times per year the interest is compounded
Arithmetic sequence (adding or subtracting)
a(n) = a1 + (n-1)d
a1: the first term
d: common difference
Arithmetic series (sum of arithmetic sequence)
S(n) = n(a1 + a(n))/ 2 or S(n) = n[2a1 +(n-1)d]/ 2 a1: the first term a(n): nth term d: common difference S(n): the sum of the first n terms
Geometric sequence
A(n) = a1 x r^(n-1)
r: common ratio; multiplier
A(n): the nth term
Geometric series (sum of geometric sequence)
S(n) = (a1-a1r^n)/ (1-r) or S(n) = (a1-a(n)r)/ (1-r) a1: first term a(n): last term in the series r: common ratio (r doesn't equal zero) S(n): sum of the first n terms
Infinite geometric series
S = a1/ (1-r)
a1: the first term
r: common ratio. -1< r<1
Pascal’s triangle
1 1 1 1 2 1 1 3. 3. 1 1. 4. 6. 4. 1 1. 5. 10. 10. 5. 1 1. 6. 15. 20. 15. 6. 1 1. 7. 21. 35. 35. 21. 7. 1
Permutations
nPr = n!/ (n-r)!
Order matters
Combinations
nCr = n!/ r!(n-r)!
Probability of inclusive events
P(A or B) = P(A) +P(B) - P(A and B)
Probability of mutually exclusive events
P(A and B) = P(A) + P(B)
Quartiles
Numbers that separate the values into four equal parts
Outliers
Upper quartile + 1.5(interquartile range)
or
Lower quartile - 1.5(interquartile range)
Probability of dependent events
P(A and B) = P(A) x P(B following A)
Probability of independent events
P(A and B) = P(A) x P(B)
Odds in favor of an event
Number of favorable outcomes/
Number of unfavorable outcomes
Odds against an event
Number of unfavorable outcomes/ number of favorable outcomes
Area of a triangle
A = 1/2 a x b x sinC
Law of sines
SinA/ a = sinB/ b
Law of cosines
a^2 = b^2 + c^2 - 2bc cosA
