Semester 1 Midterm!! Flashcards
N
Natural Numbers
Z
Integers
Q
Rational Numbers
Qc
Irrational Numbers
R
Real Numbers
C
Complex Numbers
∈
is an element of, or belongs to
ex: a ∈ A
∃
there exists
∀
for all or for every
: or |
such that
->
an implication or mapping (when used with functions)
ex: f : A -> B is a function that maps or related elements of set A to those in set B
Set
a well-defined collection of elements
[a, b]
a ≤ x ≤ b
(a, b)
a < x < b
∪
or statement (all the elements are considered in any of the listed sets)
Closure Addition
x + y ∈ R
Commutative Addition
x + y = y + x
Associative Addition
(x + y) + z = x + (y + z)
Identity Addition
x + 0 = x
Inverse Addition
x + (-x) = 0
Closure Multiplication
x * y ∈ R
Commutative Multiplication
x * y = y * x
Associative Rules
(xy) z = x (yz)
Identity Multiplication
x * 1 = x
Inverse Multiplication
x * 1/x = 1, for x ≠ 0
slope-intercept form
y = mx + b
point-slope form
y - y1 = m(x - x1)
standard form
Ax + By = C
horizontal shift
g(x) = f(x + k)
k is positive: left shift
k is negative: right shift
vertical shift
g(x) = f(x) + k
k is positive: shift up
k is negative: shift down
reflection about y-axis
g(x) = f(-x)
reflection about x-axis
g(x) = -f(x)
vertical stretch or compression
g(x) = kf(x)
k > 1: stretch
0 < k < 1: compressed
k < 0: combination of vertical or combination, along with a vertical reflection
find inverse
solve for x
put the y in the solution as an x
set as y = that thing you just did
ex:
y = 3x + 5
y - 5 = 3x
x = y-5/3
inverse function = x-5/3
dotted line
points on line are NOT part of solution set
solid line
points on line ARE part of solution set
y <
shade below line
y >
shade above line
using matrices to solve systems
A * C = B
[(x1)^2 x1 1] * a = y1
[(x2)^2 x2 1] * b = y2
[(x3)^2 x3 1] * c = y3
C = A^-1 * B
how to find determinant
A = ad - bc
inverse of 2x2
1/det * [d -b]
[-c a]
solve for matrix B
B = A^-1 * C
(A * B = C)
Cramer’s Rule
{ax + by = e
{cx + dy = f
[a b] [x] = [e]
[c d] [y] = [f]
CRAMERS RULE:
x = Dx/D, y = Dy/D
D = (ad) - (bc)
Dx = (de) - (bf)
Dy = (af) - (ce)
quadratic equation
x = (-b +- √b^2-4ac)/2a
vertex
-b/2a
vertex form
y = a(x - h)^2 + k
vertical and horizontal translations
put in vertex form
h represents horizontal shift
k represents vertical shift
reflection and dilation
if a is negative, parabola is reflected across x-axis
if |a| > 1, parabola is stretched vertically
is 0 < |a| < 1, parabola is compressed vertically
discriminant
b^2 - 4ac
if d > 0: 2 roots
if d = 0: 1 root
if d < 0: no roots
focal length
1/4a
focal point
(h, k + a)
directrix
y = k - p
p
distance from the vertex to the focus
modulus
√(a² + b²)
find quadratic function from complex roots
convert (x - (a + bi))(x - (a - bi))
into y = ax^2 + bx +c
