Algebraic Formula Sheet
Algebraic Formula Sheet
Arithmetic Operations
ac + bc = c(a + b)
b ab a=
cc
a
b
a
=
c bc
a ac
=
b
b
c
Properties of Inequalities
If a < b then a + c < b + c and a - c < b - c ab
If a < b and c > 0 then ac < bc and < cc ab
If a < b and c < 0 then ac > bc and > cc
a c ad + bc +=
b d bd
a c ad - bc -=
c d bd
a-b b-a =
c-d d-c
a+b a b =+
c cc
a
ab + ac = b + c, a = 0
a
b ad
=
c
bc
d
Properties of Exponents
xnxm = xn+m
x0 = 1, x = 0
Properties of Absolute Value x if x 0
|x| = -x if x < 0
|x| 0
| - x| = |x|
|xy| = |x||y|
x |x| =
y |y|
|x + y| |x| + |y| Triangle Inequality |x - y| |x| - |y| Reverse Triangle Inequality
(xn)m = xnm
(xy)n = xnyn
n
xm =
1n
xm =
xn
1 m
-n
n
x
y
yn
=
=
y
x
xn
n
x
xn
y = yn
1 = xn x-n xn = xn-m xm
x-n
=
1 xn
Distance Formula Given two points, PA = (x1, y1) and PB = (x2, y2), the distance between the two can be found by:
d(PA, PB) = (x2 - x1)2 + (y2 - y1)2
Number Classifications Natural Numbers : N={1, 2, 3, 4, 5, . . .}
Whole Numbers : {0, 1, 2, 3, 4, 5, . . .}
Properties of Radicals
nx
=
1
xn
x nx n = y ny
n xy = n x n y
n xn = x, if n is odd
m
nx=
x mn
n xn = |x|, if n is even
Integers : Z={... ,-3, -2, -1, 0, 1, 2, 3, .. .}
Rationals : Q= All numbers that can be written as a fraction with an integer numerator and a
a nonzero integer denominator,
b Irrationals : {All numbers that cannot be expressed as the ratio of two integers, for example
5, 27, and }
Real Numbers : R={All numbers that are either a rational or an irrational number}
1
Logarithms and Log Properties
Definition y = logb x is equivalent to x = by Example log2 16 = 4 because 24 = 16 Special Logarithms ln x = loge x natural log where e=2.718281828... log x = log10 x common log
xa + xb = x(a + b) x2 - y2 = (x + y)(x - y) x2 + 2xy + y2 = (x + y)2 x2 - 2xy + y2 = (x - y)2 x3 + 3x2y + 3xy2 + y3 = (x + y)3 x3 - 3x2y + 3xy2 - y3 = (x - y)3
Logarithm Properties
logb b = 1 logb bx = x ln ex = x
logb 1 = 0 blogb x = x eln x = x
logb (xk) = k logb x logb (xy) = logb x + logb y
x logb y = logb x - logb y
Factoring x3 + y3 = (x + y) x2 - xy + y2 x3 - y3 = (x - y) x2 + xy + y2 x2n - y2n = (xn - yn) (xn + yn) If n is odd then, xn - yn = (x - y) xn-1 + xn-2y + ... + yn-1 xn + yn = (x + y) xn-1 - xn-2y + xn-3y2... - yn-1
Linear Functions and Formulas
Examples of Linear Functions
y
y
y=x x
y=1 x
linear f unction 2
constant f unction
Constant Function
This graph is a horizontal line passing through the points (x, c) with slope m = 0 :
y = c or f (x) = c
Linear Function/Slope-intercept form
This graph is a line with slope m and y - intercept(0, b) :
y = mx + b or f (x) = mx + b
Slope (a.k.a Rate of Change)
The slope m of the line passing through the points (x1, y1) and (x2, y2) is : m = y = y2 - y1 = rise
x x2 - x1 run
Point-Slope form
The equation of the line passing through the point (x1, y1) with slope m is :
y = m(x - x1) + y1
Quadratic Functions and Formulas
Examples of Quadratic Functions
y
y
y = x2 x
y = -x2 x
parabola opening up
parabola opening down
Forms of Quadratic Functions
Standard Form
Vertex Form
y = ax2 + bx + c
or f (x) = ax2 + bx + c
y = a(x - h)2 + k
or f (x) = a(x - h)2 + k
This graph is a parabola that opens up if a > 0 or down if
a < 0 and has a vertex at
b
b
- ,f -
.
2a
2a
This graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h, k).
3
Quadratics and Solving for x
Quadratic Formula To solve ax2 + bx + c = 0, a = 0,
use :
-b ? b2 - 4ac
x=
.
2a
The Discriminant
The discriminant is the part of the quadratic equation under the radical, b2 - 4ac. We use
the discriminant to determine the number of real solutions of ax2 + bx + c = 0 as such :
Square Root Property
Let k be a nonnegative number. Then the solutions to the equation
x2 = k are given by x = ? k.
1. If b2 - 4ac > 0, there are two real solutions. 2. If b2 - 4ac = 0, there is one real solution. 3. If b2 - 4ac < 0, there are no real solutions.
Other Useful Formulas
Compound Interest
r nt A=P 1+
n where: P= principal of P dollars r= Interest rate (expressed in decimal form) n= number of times compounded per year t= time
Continuously Compounded Interest
A = P ert
where: P= principal of P dollars r= Interest rate (expressed in decimal form) t= time
Circle
(x - h)2 + (y - k)2 = r2
This graph is a circle with radius r and center (h, k).
Ellipse
(x - h)2 (y - k)2 a2 + b2 = 1
This graph is an ellipse with center (h, k) with vertices a units right/left from the center and vertices b units up/down from the center.
Hyperbola
(x - h)2 (y - k)2
-
=1
a2
b2
This graph is a hyperbola that opens
left and right, has center (h, k), vertices
(h ? a, k); foci (h ? c, k), where c comes from c2 = a2 + b2 and
asymptotes that pass through the center b
y = ? (x - h) + k. a
(y - k)2 (x - h)2
-
=1
a2
b2
This graph is a hyperbola that opens up and down, has center (h, k), vertices (h, k ? a); foci (h, k ? c), where c comes from c2 = a2 + b2 and asymptotes that pass through the center
a y = ? (x - h) + k.
b
Pythagorean Theorem
A triangle with legs a and b and hypotenuse c is a right triangle if and only if
a2 + b2 = c2
4
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