Algebra II Final Study Guide
How do you read -> in the end behavior of graphs?
"approaches"
x-intercepts of intercept form
"the OPPOSITE" of p and q
What is the completion of the perfect square trinomial?
(b/2)^2
Vertex of vertex form
(h,k) *(x-h) h=1 NOT -1 because set = to 0 h moves horizontally k moves vertically
Vertex of vertex form of a quadratic equation
(h,k) NOT -h
How to graph a standard form parabola
*follow the rules 1) find the axis of symmetry 2) plug in this x to find y 3) graph the vertex 4) graph the y-intercept and the point symmetrical to it 5) if you want plug in any x to get a y and graph 6) connect the dots *check rules 1 and 2
b^2-4ac=0
1 real solution
Rules of standard form
1) a>0 graph opens upward a<0 graph opens down 2) Narrow graph if the absolute value of a>1 3) axis of symmetry x=-b/2a 4) vertex=(-b/2a, y)- you must plug in x to find y 5) y-intercept= c value (0,c)
Factoring: Perfect square trinomial
1) a^2+2ab+b^2=(a+b)^2 2) a^2-2ab+b^2=(a-b)^2
Square roots rules
1) always simplify as much as possible 2) no roots in the denominator simplified when: no radicand has a perfect square as a factor other than 1 and there are no radicands in the denominator
Graph vertex form
1) find the opposite of h 2) plug in to find y 3) graph vertex 4) find points on opposite sides 5) connect the dots
Graph intercept form
1) find the opposites of p and q and graph 2) use x=p+q/2 to find the axis of symmetry 3) plug in to find y 4) graph the vertex 5) connect the dots
Graphing a system of inequalities
1) graph each inequality separately 2) the answer is where the shading overlaps
Graphing and solving quadratic inequalities
1) graph the equation 2) pick a test point (easiest as long as graph isn't going through it is (0,0) 3) false= shaded where test point is NOT true= shaded where test point is *shade where test point is!
Completing the square
1) move to other side 2) add (b/2)^2 to both sides 3) do the math 4) factor 5) find the square root 6) simplify
Solving quadratic inequalities algebraically
1) solve the equation 2) mark your points on a number line 3) use a test point (0) < or ≤ = <x< > or ≥ = x< or x> < > = open circle ≤ ≥= closed circle
Standard form of complex numbers with fractions
11/26 + (23/26)i
b^2-4ac<0
2 imaginary solutions
b^2-4ac>0
2 real solutions
Pure imaginary vs. complex numbers
3i= pure imaginary 3+4i= complex number
Application problems for quadratic graphs
???
Writing a quadratic equation in vertex form
???
Polynomial
A monomial or the sum of monomials
Scientific Notation Rules
C*10^n 1≤C<10 this means that there can only be one whole number (it must be less than 10 and more than 1) in the answer and ten must be multiplied to however many places it is necessary to move the decimal to get the true number
a^-m
Negative exponent take the reciprocal a CANNOT equal 0 because 0 can't be in the denominator 1/a^m
(a^m)^n
Power of a Power multiply the exponents a^m*n
(ab)^m
Power of a product make sure you distribute it to everything in the parentheses a^m * b^m
(a/b)^m
Power of a quotient make sure you distribute it to everything in the parentheses a^m/b^m
a^m * a^n
Product of powers add the exponents a^m+n
a^m/a^n
Quotient of exponents subtract the exponents a^m-n
a^0
Zero Exponent anything to the power 0 = 1 1
Square roots
a number is a square root of a number if r^2=s s must be positive √s= two roots √s and -√s
Monomial
a number, a variable, or the product of a number and a variable (it is also a polynomial) -there cannot be any fractional exponents or any variables in the denominator (this includes negative exponents) there also cannot be a variable under a positive or negative square root sign
Complex conjugate of a+bi
a-bi
*Know factoring
a=1 a does NOT=1 common factors *highest exponent= the number of solutions
Factoring: Difference of squares
a^2 -b^2 = (a+b)(a-b)
Rationalizing the denominator: (a+√b)(a-√b)=
a^2-b
Function
an operation where every x has one y
Discriminant
b^2-4ac
Subtracting polynomials
change the signs of the terms being subtracted, then add ex: a-b = a+(-b)
Adding polynomials
combine all like terms
Completing the square when a does not = 0
divide everything by a
Axis of symmetry
divides the parabola into mirror images and passes through the vertex it is an equation because it's a line in the parent function x=0 x-coordinate of the vertex quadratic equations: -b/2a
> and < type of line on graph
dotted
Monomial
expression that is either a number, a variable, or a product of a number and one or more variables
End behavior of graphs when the degree is odd and the leading coefficient is positive
f(x)-> +∞ as x-> +∞ f(x)-> -∞ as x-> -∞ so as x gets bigger y will get bigger and as x gets smaller y will get smaller (the ∞ signs are the same: +∞ +∞ vs. -∞ -∞)
End behavior of graphs when the degree is even and the leading coefficient is positive
f(x)-> +∞ as x-> -∞ f(x)-> +∞ as x-> +∞ so as x gets smaller y will get bigger and as x gets bigger y will also get bigger (the ∞ signs always start with +∞ and the end is different: +∞ -∞ vs. +∞ +∞)
End behavior of graphs when the degree is odd and the leading coefficient is negative
f(x)-> +∞ as x-> -∞ f(x)-> -∞ as x-> +∞ so as x gets smaller y will get bigger and vise versa (the ∞ signs are opposite: +∞ -∞ vs. -∞ +∞)
End behavior of graphs when the degree is even and the leading coefficient is negative
f(x)-> -∞ as x-> +∞ f(x)-> -∞ as x-> -∞ so as x gets bigger y will get smaller and as x gets smaller y will also get smaller (the ∞ signs always start with -∞ and the end is different: -∞ +∞ vs. -∞ -∞)
Parent function of standard form
f(x)/g(x)/q(x)/p(x)/etc.=x^2 vertex=(0,0)
How to tell the difference between graph and algebraic inequalities
graph has y on one side algebraic has 0 on one side
Application problem: Modeling launched objects
h=-16t^2+ho (dropped) h=-16t^2+vot+ho (launched or thrown) h=where it lands ho=initial height v=initial vertical velocity in ft/sec v>0 launched upward v<0 launched downward v=0 not launched -solving for t
Application problem: Dropping objects
h=-16t^2+hₒ hₒ= initial height when h=o object is hitting the ground t=time (usually finding this) -don't have to put +- because will never be negative seconds
Axis of symmetry of intercept form
halfway between (p,0) & (q,0) formula is (p + q)/2
Solving quadratic equations: Zero Product Property
if the product of 2 expressions is 0 then one or both of the equations equal 0
i
imaginary unit √-1 i^2=-1
Complex numbers
in the real number system there is no solution for x=√-1
Rationalizing the denominator: a-√b
multiply the numerator and denominator by a+√b (conjugate)
Rationalizing the denominator: a+√b
multiply the numerator and denominator by a-√b (conjugate)
Rationalizing the denominator: √b
multiply the numerator and denominator by √b
Multiplication of complex numbers
normal multiplication but must remember that i^2=-1
Positive root=
principle root
How to find the end behavior of graphs on a graphing calculator
put in the function look at the graph decide what points to look at on the table go to the table graph the points on the table check to make sure it looks like the calculator's graph
What is a polynomial function to the 4th degree called?
quartic (this is the type)
What is a polynomial function to the 5th degree called?
quintic (this is the type)
Complex number system
real numbers imaginary numbers standard form: a+bi write with the real number first
≥ and ≤ type of line on graph
solid
All quadratic forms
standard- y=ax^2+bx+c (x=-b/2a -> plug in get y) *vertex- y=a(x-h)^2+k (x=(h,k) -opposite of h) *intercept- y=a(x-p)(x-q) (x=p+q/2 ->plug in to get y -p and q are the x-intercepts) *can't get y-intercept
Evaluating by Direct Substitution
substitute the value of x that is given anywhere there is an x in the equation
Polynomial function rules
terms written in descending order of exponents standard form shown in the diagram a(sub)n- the leading coefficient (does NOT eqaul 0/ tells which x^n it goes with so no confustion) n- the degree of the polynomial a(sub)0- constant term the exponent must be a whole number and positive the coefficient must be a real number
Quadratic Graphs: If a<0 in standard and vertex form
the graph opens downward and the vertex's y-value is the maximum value
Quadratic Graphs: If a>0 in standard and vertex form
the graph opens upward and the vertex's y-value is the minimum value
Radicand
the number under the radical
How is -x^2 read?
the opposite of x squared
Binomial
the sum of 2 monomials
Trinomial
the sum of 3 monomials
What is necessary for any of the exponent rules to be true?
they must have the same bases
Polynomial function to the 0 degree
type: constant standard form: f(x)= a(sub)0
Polynomial function to the 3 degree
type: cubic standard form: f(x)= a(sub)3x^3 + a(sub)2x^2 + a(sub)1x + a(sub)0
Polynomial function to the 1 degree
type: linear standard form: f(x)= a(sub)1x + a(sub)0
Polynomial function to the 2 degree
type: quadratic standard form: f(x)= a(sub)2x^2 + a(sub)1x + a(sub)0
Polynomial function to the 4 degree
type: quartic standard form: f(x)= a(sub)4x^4 + a(sub)3x^3 + a(sub)2x^2 + a(sub)1x + a(sub)0
Converting to standard form
vertex: 1) square (x-h) 2) multiply by a 3) simplify intercept: 1) multiply (x-p) and (x-q) 2) multiply by a 3) simplify
End behavior of Graphs
what the graph of the polynomial does when the end part is not on the graph
Multiplying polynomials
when vertically do like a normal multiplication problem (make sure you put the =/- sing in front of the number so you know whether to add or subtract it from the next one also make sure to line up like terms- you may have to skip some spaces on the right after a couple of times just like in regular multiplication) when horizontally use the distributive property (every term in the first must multiply all terms in the second)
Finding the zeros/roots of
where y=0 factoring but set equal to zero
Graphing complex numbers
x-axis is, instead, going to be the real part of the complex number, so we call it the real axis. That makes the y-axis our imaginary axis and will represent the imaginary part of the number. so graphing -2 + 5i, is basically, the same as graphing (-2,5) on a normal x/y-axis the origin is (0,0i)
How should you write the answer of a completing the square problem?
x=3/2+-√37/2
Perfect square trinomial
x^2+2xy+y^2
Vertex form of a quadratic equation
y=a(x-h)^2+k
Vertex form
y=a(x-h)^2+k *graph is the parent function
Intercept form
y=a(x-p)(x-q)
standard form of a parabola
y=ax^2+bx+c red dot= vertex quadratic graph (parabola)
Radical
√ (just the symbol)
Simplifying radicals
√12=√4*√3=2√3
Product property of square roots
√a * √b=√ab
Quotient property of radicals
√a/b=√a/√b
The absolute value of a+bi
√a^2+b^2 (all is under the root)