Algebra Flashcards
DOTS
difference of two squares
a^2 - b^2 = (a+b)(a-b)
SODOTC
sum or difference of two cubes
a^3 _ b^3 = (a _ b) (a^2 _ ab + b^2)
SIGNS : SOAP
same, opposite, always positive
a^3 + b^3 = (a + b) (a^2 - ab + b^2)
a^3 - b^3 = (a - b) (a^2 + ab + b^2)
completing the square
- x^2 and y^2 coeffiecent = 1
- divide coefficient of middle term by 2
- square answer
- add product as constant of equation
can be used in converting circles from general to standard form
slope-intercept form
linear function
y=mx+b
degree of x is 1
m = slope
b = y-intercept
general form
linear function
Ax + By + C = 0
degree of x is 1
y-intercept (from general form)
linear function
b = -C / B
slope of a line equation (from general/standard form)
linear function
slope = -A / B
slope of a line equation
linear function
slope = change in y / change in x
slope = y2 - y1 / x2 - x1
domain and range
linear function
domain = R
range = R
all real numbers
point-slope formula
linear function
y - y1 = m(x - x1)
given point and slope
get m by using slope formula
general form
rational function
f(x) = P(x) / Q(x)
contains variables in denominator
domain
rational function
domain = {x|x ≠ 0}
all real nos. EXCEPT value that will make denominator = 0
range
rational function
- interchange x and y in equation
- solve for y
domain
odd root radical function
domain = R
all real numbers
domain
even root radical function
- solve ( radicand ≥ 0 )
radicand should never be negative (imaginary number)
standard form
quadratic function
y = ax^2 + bx + c
highest degree of x is 2
x-coordinate of vertex (from standard form)
quadratic functions
x = -b / 2a
y-coordinate of vertex (from x-coordinate of standard form)
quadratic functions
replace x in the equation with answer from x = -b / 2a
y-coordinate of vertex (from standard form)
quadratic functions
k = 4ac - b^2 /
4a
vertex form
quadratic function
y = a(x-h)^2 + k
given vertex of the parabola
domain
quadratic function
domain = R
range
quadratic function
R: {y|y _ k}
k : y-coordinate of vertex
if parabola opens UPWARD (+a), greater than or equal to
if parabola opens DOWNWARD (-a), less than or equal to
circle standard form
conic sections
(x-h)^2 + (y-k)^2 = r^2
center : (h,k)
radius : r
conic sections general form
Ax^2 + Cy^2 + Dx + Ey + F = 0
A : coefficient of x^2
C : coefficient of y^2
F : constant
parabola shortcut
conic sections general form
A or C is squared
(x^2 OR y^2)
only one of the variables is squared
parabola standard form
conic sections
opening up or down
(x-h)^2 = 4a (y-k)
only the x variable is being squared
-4a : opening DOWN
+4a : opening UP
(visualize cartesian plane and y-axis)
parabola standard form
conic sections
opening left or right
(y-k)^2 = 4a (x-h)
only they variable is being squared
-4a : opening LEFT
+4a : opening RIGHT
(visualize cartesian plane and x-axis)
hyperbola shortcut
conic sections general form
A and C are squared
different signs
multiplying A and C will give a negative product
hyperbola standard form
horizontal transverse axis
[(x-h)^2 / a^2] - [(y-k)^2 / b^2] = 1
x is first term
hyperbola standard form
vertical transverse axis
[(y-k)^2 / a^2] - [(x-h)^2 / b^2] = 1
y is first term
circle shortcut
conic sections general form
A and C are squared
A and C are equal
coefficients of x^2 and y^2 are equal
ellipse shortcut
conic sections general form
A and C are squared
same signs, but not equal
A and C are not the same number
A x C will give positive product
ellipse standard form
conic sections
[(x-h)^2 / a^2] + [(y-k)^2 / b^2] = 1
if b > a = vertical major axis
if a > b = horizontal major axis
larger number is below X = HORIZONTAL
larger number is below Y = VERTICAL
center of a circle
general form ( if A and C = 1 )
h = -D / 2
k= -E / 2
r = -F + h^2 + k^2
D : coefficient of X
E : coefficient of Y
F : constant
converting logarithm -> exponential form
BREIN
Base Raised to Equal Is Natira
logbA = e -> b^e = A
exponential solving shortcut
BEBE Form
B^e = B^e
- make bases the same
- equate exponents to each other
rationalizing
denominator with radical
multiply conjugate of denominator
square root goal : DOTS
cube root goal : SODOTC
cube root conjugate : a^2 _ ab + b^2
- first sign is opposite of given sign
