Algebra II Vocab
Perfect Square Trinomials
(a+b)^2 = a^2 + 2ab + b^2
Graphing simple rational functions
1. Find Domain and Range 2. Find Asymptotes 3. Graph it
Adding and Subtracting Rational Expressions
1. Find Least common denominator 2. simplify faction
Checking for extraneous solutions
1. Isolate root 2. Squaring Both Sides 3. Plug it in the equation to see if it works
How to divide Polynomials
1. Long division 2. Synthetic division
Solve by Factoring process
1. Move all terms to one side of the equation, usually the left, using addition or subtraction. 2. Factor the equation completely. 3. Set each factor equal to zero, and solve. 4. List each solution from Step 3 as a solution to the original equation.
Multiplying Rational Expressions
1. Simplify faction 2. Multiply fraction
Properties of Exponents
1. x^n * n^m = x^(n+m) 2. (x^n)/(x^m) = x^(n-m) 3. (x^n)^m = x^(n*m) 4. (x*y)^n = x^n * y^n 5. (x/y)^n = (x^n)/(y^n)
Length of Latus Rectum for Ellipse
2b^2/a
Solving absolute value inequalities
2|3x+9|<36 2|3x+9|2<362 |3x+9|<18 −18<3x+9<18 −18−9<3x+9−9<18−9 −27<3x<9 −273<3x3<93 −9<x<3
complex numbers
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x² = −1. Because no real number satisfies this equation, i is called an imaginary number.
Rational Expression
A fraction in which the numerator and/or the denominator are polynomials.
Interval Notation
A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Ex. Let's look at the intervals we did with the set-builder notation: 0 is less than or equal to x < 4 x > -1 x is less than or equal to 2 x < 0 or x > 3 Let's start with the first one: 0 is less than or equal to x < 4 This is what it means; number line showing x is greater than or equal to 0 and less than 4 So, we write it like this: [ 0, 4 ) This is interval notation! Use [ or ] for closed dots less than or equal to, greater than or equal to Use ( or ) for open dots < , >
Factor Theorem
As the Remainder Theorem points out, if you divide a polynomial p(x) by a factor x - a of that polynomial, then you will get a zero remainder. Let's look again at that Division Algorithm expression of the polynomial: ADVERTISEMENT p(x) = (x - a)q(x) + r(x) If x - a is indeed a factor of p(x), then the remainder after division by x - a will be zero. That is: p(x) = (x - a)q(x) In terms of the Remainder Theorem, this means that, if x - a is a factor of p(x), then the remainder, when we do synthetic division by x = a, will be zero.
Standard Form
Ax + By + C = 0 (A and B cannot both be 0)
Leading Coefficient Test for polynomials
F(x) = ax^n if a > 0 and n = even the graph will increase without bound positively at both endpoints. A good example of this is the graph of x^2. if a > 0 and n = odd the graph will increase without bound positively at the right end and decrease without bound at the left end. A good example of this is the graph of x^ 3. if a < 0 and n = even the graph will decrease without bound positively at both endpoints. A good example of this is the graph of -x^2. if a < 0 and n = odd the graph will decrease without bound positively at the right end and increase without bound at the left end. A good example of this is the graph of -x^3.
How to find the inverse of a function
Find the domain and range of the function y=1x+3−5 . To find the excluded value in the domain of the function, equate the denominator to zero and solve for x . x+3=0⇒x=−3 So, the domain of the function is set of real numbers except −3 . The range of the function is same as the domain of the inverse function. So, to find the range define the inverse of the function. Interchange the x and y . x=1y+3−5 Solving for y you get, x+5=1y+3⇒y+3=1x+5⇒y=1x+5−3 So, the inverse function is f−1(x)=1x+5−3 .
How to find the Domain of fuction
Find the domain of the function y=1x+3−5 . To find the excluded value in the domain of the function, equate the denominator to zero and solve for x . x+3=0⇒x=−3 So, the domain of the function is set of real numbers except −3 .
How to find Asymptotes of function
Find the vertical and horizontal asymptotes of the function f(x)=5x−1 . To find the vertical asymptote, equate the denominator to zero and solve for x . x−1=0⇒x=1 So, the vertical asymptote is x=1 Since the degree of the polynomial in the numerator is less than that of the denominator, the horizontal asymptote is y=0 .
How to find the range of the function
Function = y = (1/(x+3))−5 The range of the function is same as the domain of the inverse function. So, to find the range define the inverse of the function. Interchange the x and y . So, x=(1/(y+3))−5 Then solve for y
Function Notation
Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a rather lengthy written explanation. Ex. (Basic) f(x) = 3x+1
The Horizontal Line Test
If the horizontal line intersects the graph of a function in all places at exactly one point, then the given function should have an inverse that is also a function. We say this function passes the horizontal line test.
The Vertical Line Test
If you can draw a vertical line anywhere on a graph so that it hits the graph in more than one spot, then the graph is NOT a function.
How to find inverse function graph
If you know the graph of the regular function then reflect it over the line y = x.
Multiplying Polynomials with More Than One Variable
Multiply. (4x - 7xy)(2y + 3x) 4x • 2y = 8xy 4x • 3x = 12x^2 −7xy • 2y = −14xy^2 −7xy • 3x = −21x^2y 8xy + 12x^2 - 14xy^2 - 21x^2y Answer The product is 8xy + 12x2 - 14xy^2 - 21x^2y.
Adding Polynomials with More Than One Variable
Problem Add. (4x2 - 12xy + 9y2) + (25x2 + 4xy - 32y2) 4x2 +(−12xy) + 9y2 + 25x2 + 4xy + (−32y2) (4x2 +25x2) +[(−12xy)+ 4xy] + [9y2+ (−32y2)] 29x2 + (−8xy) +(−23y2) Answer The sum is 29x2 - 8xy - 23y2.
Subtracting Polynomials with More Than One Variable
Problem Subtract. (14x3y2 - 5xy + 14y) - (7x3y2 - 8xy + 10y) 14x3y2 - 5xy + 14y - 7x3y2 + 8xy - 10y 14x3y2 - 7x3y2 - 5xy + 8xy + 14y - 10y 7x3y2 + 3xy + 4y Answer The difference is 7x3y2 + 3xy + 4y
Dividing Polynomials with More Than One Variable
Same thing as multiplying
Long Division of Polynomials
Same thing as regular long division
Dividing Rational Expressions
Similar to multiplying
parent quadratic function
The "Parent" Graph: The simplest parabola is y = x2, whose graph is shown at the right. The graph passes through the origin (0,0), and is contained in Quadrants I and II. This graph is known as the "Parent Function" for parabolas, or quadratic functions.
Remiander Theorem
The Remainder Theorem starts with an unnamed polynomial p(x), where "p(x)" just means "some polynomial p whose variable is x". Then the Theorem talks about dividing that polynomial by some linear factor x - a, where a is just some number. Then, as a result of the long polynomial division, you end up with some polynomial answer q(x) (the "q" standing for "the quotient polynomial") and some polynomial remainder r(x).
Domain and Range of Rational Functions
The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.
Absolute Value Function in Equation Form
The general form of an absolute value function that is linear is y = |mx + b| + c where the vertex (low or high point) is located at (-b/m, c) the vertical line: -b/m divides the graph into two equal halves
Fundamental Theorem of Algebra
The highest degree of a polynomial is the max amount of roots
Composite Function Notation
The result of f() is sent through g() It is written: (g º f)(x) Which means: g(f(x))
Standard Complex Form
a + bi
Definition of a Function
a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.
Difference of Squares
a^2 - b^2 = (a+b)(a-b)
discriminant to determine number and nature of solutions
discriminant = b^2−4ac If sum = 1 then there are 2 solutions If sum = 0 then there are 1 solutions If sum = -1 then there are 0 solutions
Vertex Form of quadratic
f (x) = a(x - h)2 + k (h, k) = vertex
Rational Roots Theorem
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Solve Quadratic Equations by Completing the square
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Solve Quadratic Equations by Square Root Method
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Solving absolute value equations
o solve an absolute value equation as |x+7|=14 You begin by making it into two separate equations and then solving them separately. x+7=14 x+7−7=14−7 x=7 or x+7=−14 x+7−7=−14−7 x=−21
Solve by quadratic formula
x= (−b± sqrt(b^2−4ac))/2a ax^2 + bx + c = 0
Slope-Intercept Form (Linear)
y = mx + b, m = slope, b = y intercept Ex. (Picture)
Point-Slope Form
y − y1 = m(x − x1), m = slope, (x1,y1) = point on line
