Section 1
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Cards (460)
Section 1
(50 cards)
Cot=
X/y reciprocal of tan y/x
The Change-of-Base Property: Introducing Common and Natural Logarithms
Introducing Common Logarithms Introducing Natural Logarithms
The Product Rule
Let b, M, and N be positive real numbers with b 1. The logarithm of a product is the sum of the logarithms.
Tangent
O/A or Y/x
60°
Pi/3
sin ø=?
cos (90-ø)
Condensing Logarithmic Expressions
sin 72
cos 18
Sine=
O/H, or y
0° or 360°
2pi
30°
Pi/6
The Power Rule
Let b and M be positive real numbers with b 1, and let p be any real number. The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.
If terminal rotates clockwise
Produces a negative angle
Complimentary angles
Add up to 90° if it is greater than pi/2, it has no compliment
The Quotient Rule
Let b, M, and N be positive real numbers with b 1. The logarithm of a quotient is the difference of the logarithms.
Tan and cot are (even or odd?)
Odd
90°
Pi/2
Periodic function
Continues forever form, kind of like a wavelength looks. Like, copy and paste. Can be written as f(t+c) = f(t) where c is a real positive # sin(t+2πn)
cos ø=?
sin (90-ø)
Circumference
2pir
The domain of cos
All real numbers
Radian
Used as a new way to measure angles in terms of pi.
Properties of Common Logarithms
Inverse Properties of Logarithms
For b > 0 and b 1, log base b, b∧x = x b∧log bx = x
The Domain of a Logarithmic Function
The domain of an exponential function of the form f(x) = b∧x includes all real numbers and its range is the set of positive real numbers. Because the logarithmic function reverses the domain and the range of the exponential function, the domain of a logarithmic function of the form: log base b, x :is the set of all positive real numbers. In general, the domain of: log base b, x: consists of all x for which g(x) > 0.
Degree to radian
Degree • pi/180
Cos (even or odd?)
Even
Coterminal angles
Angles in the same position
Csc=
1/y or the reciprocal of y(sin)
The domain of sin
All real numbers
Sec=
1/x or the reciprocal of x (cos)
Sine (even or odd?)
Odd
Natural Logarithms
The logarithmic function with base e is called the natural logarithmic function. f(x) = log base e, x The function f(x) = lnx is usually expressed
sin 10
cos 80
Characteristics of Logarithmic functions of the Form f(x) = log base b, x
Common Logarithms
The logarithmic function with base 10 is called the common logarithmic function. f(x) = log₁₀ x The function f(x) = log x :is usually expressed
Range of cos
Between -1>x< 1 shit be happening "equal to -1 & 1 as well"
cos 70
sin 20
Properties of Natural Logarithms
Radian to degree
Radian measurement • 180/pi
Half rotation
180° or pi
45°
Pi/4
Definition of the Logarithmic Function
Definition of the Logarithmic Function
The Change-of-Base Property
For any logarithmic bases a and b, and any positive number M, The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.
If terminal rotates counter clockwise
Produces a positive angle
Cosine=
A/H, or x
Range of sin
-1>y<1 is where the shit goes down "less than or equal to"
Basic Logarithmic Properties Involving One
1. logbb = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b) 2. logb1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1)
Full rotation
360° or 2pi
Supplementary angles
Adds to 180°. If an angle is greater than pi, it has no supplement
Section 2
(50 cards)
special right triangles
there are two special right triangles: 30-60-90 and 45-45-90
Phase Shift
bx+/-c=0 bx+/-c=2π
cosθ=
x/r
Cos (and sec) is positive in quadrants?
I and IV
Domain
ℝ (-∞, ∞)
a > 1
vertical stretch
Area of a Sector
A=1/2(r^2)θ θ must be in radians
0⁰
0 (1,0)
Even Functions When Neg.
cos(-θ)=cosθ sec(-θ)=secθ they're the same as the non-negated outcome so you should treat them as such
b is negative
vertical flip across y-axis (in cosine, no visual change)
r=
√(x²+y²)
b < 1
less cycles
y = A sin(Bx) + C
A: amplitude (vertical stretch/compression C: midline (vertical shift) B= 2π/ period (horizontal stretch/compression)
horizontal shift
(x = h)
cotθ=
x/y, y≠0
Period
2π/|b|
Reference angle θ'
The acute angle formed by the terminal side of the angle and the horizontal axis.
Tan (and cot) is positive in quadrants?
I and III
Cosine Curve
Starts at amplitude
unit circle
k is positve
shift up
60⁰
π/3 (1/2, √3/2)
Arc Length
S=rθ θ must be in radians
Sin (and csc) is positive in quadrants?
I and II
a < 1
vertical shrink
Unit Circle
x^2+y^2=1 points on circle must satisfy equation
secθ=
r/x, x≠0
If θ is in Quadrant I, then θ' is
θ'=θ
a is negative
horizontal flip on x-axis
45⁰
π/4 (√2/2, √2/2)
tanθ=
y/x, x≠0
90⁰
π/2 (0,1)
Sine Curve
Starts at 0
increments
period / 4
graph cos(x)
k is negative
shift down
cscθ=
r/y, y≠0
Amplitude (in terms of max & min)
(max + min) / 2
vertical shift
(y = k)
30⁰
π/6 (√3/2, 1/2)
reference angles
the acute angle formed by the terminal side of the angle and the x-axis
sinθ=
y/r
Range
[-a, a]
Odd Functions
sin(-θ)= -(sinθ) csc(-θ)= -(cscθ) tan(-θ)= -(tanθ) cot(-θ)= -(cotθ) the same as the non-negated but just negative.
b > 1
more cycles
reference right triangles
triangles formed by drawing a vertical perpendicular line from the intersection of the terminal side of an angle and the unit circle to the x-axis Distances are always positive
b
period
Increments
period/4
graph sin(x)
a
amplitude
Section 3
(50 cards)
arccos
Domain: [-1, 1] Range: [0, π]
Period of csc(t)
Period: 2π (same as sin(t))
- cos x
Domain of cot(t)
Domain: All real numbers other than nπ for any integer, n. (same as csc(t))
Solving a Right Triangle
Finding all side lengths and angle measures of a triangle.
phase shift =
c/|b|
2 cos x
- cos(x) + 1
Equation of csc(t)
t=1/y
sin x
Period of sin(t)
Period: 2π (same as cos(t))
Period of cot(t)
Period: 2π (same as csc(t))
Range of sin(t)
Range: [−1,1] (same as cos(t))
Equation of sec(t)
t=1/x
Domain of tan(t)
Domain: All real numbers other than π/2+nπ for any integer n. (same as sec(t))
- sin x
Domain of sin(t)
Domain: All reals (−∞, ∞) (same as cos(t))
Range of tan(t)
Range: All reals (−∞, ∞) (same as sec(t))
arcsin
Domain: [-1, 1] Range: [-π/2, π/2]
If θ is in Quadrant IV, then θ' is
θ'=360⁰- θ θ'= 2π - θ
Range of cos(t)
Range: [−1,1] (same as sin(t))
Range of sec(t)
Range: All reals (−∞, ∞) (same as tan(t))
Equation of cot(t)
t=x/y
Range of csc(t)
Range: (-∞,-1]∪[1,∞) (same as cot(t))
cos x
arctan
Domain: [-∞, ∞] Range: [-π/2, π/2]
Graph of csc(t)
...
- sin(x) + 2
Inverse Trigonometric Ratios
You can use sin⁻¹, cos⁻¹ and tan⁻¹ to find the measure of an angle in a right triangle given its sides.
If θ is in Quadrant II, then θ' is
θ'=180⁰- θ θ'= π - θ
arccot x
D: (-∞,∞) R: (0,π)
Period of tan(t)
Period: π (same as cot(t))
Equation of cos(t)
t=x
sin(2x)
Arctan(1)
tan(x)=1 x=π/4
Graph of sin(t)
Trigonometric Ratio
A ratio of the lengths of two sides in a right triangle to the angle.
Range of cot(t)
Range: (-∞,-1]∪[1,∞) (same as csc(t))
Equation of tan(t)
t=y/x
arccsc x
D: (⁻∞ , -1] U [1 , + ∞) R: (-π/2, 0) U (0, π/2]
Domain of csc(t)
Domain: All real numbers other than nπ for any integer, n. (same as cot(t))
Domain of sec(t)
Domain: All real numbers other than π/2+nπ for any integer n. (same as tan(t))
Domain of cos(t)
Domain: All reals (−∞, ∞) (same as sin(t))
Steps for Csc(x) and Sec(x) Graphs
1. Graph Reciprocal 2. Everywhere you see x-int draw asymptotes. 3. Draw parabolas on the maxs and mins.
arcsec x
D: (⁻∞ , -1] U [1 , + ∞) R: [0 , π/2) U (π/2 , π]
Compositon of Functions
Graph of tan(t)
...
Period of cos(t)
Period: 2π (same as sin(t))
Equation of sin(t)
t=y
If θ is in Quadrant III, then θ' is
θ'= θ - 180⁰ θ'= θ - π
Section 4
(50 cards)
cosx=
sin(∏/2 -x)
Care must be taken when squaring both sides of a trigonometric equation to obtain a quadratic because
this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous
finite sequence
a sequence whose domain consists of the first n positive integers only
D (-∞,∞) R (-π/2,π/2)
y=arctanx
The key to verifying identities is
the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions.
D (-∞,∞) R (0,π)
y=arccotx
A geometric sequence is determined by:
Its initial term and its common ratio.
sin²x + cos²s=
1
To solve an equation in which two or more trigonometric functions occur
collect all terms on one side and try to separate the functions by factoring or by using appropriate identities.
cscx=
sec(∏/2 -x)
Tricks when proving Trig Identities
Conjugate, factoring, and common denominator
Arithmetic Sequence
A sequence in which each term is found by adding a fixed amount to the previous term
Even-Odd Identities
sin(-x) = - sin x cos(-x) = cos x tan (-x) = - tan x csc (-x) = - csc x sec (-x) = sec x cot (-x) = - cot x
Common Ratio
This is denoted by r. It is the number that we always multiply the previous term by to obtain the following term. r = a2/a1= a3/a2
Describe a strategy for verifying the identity 2 2 sin (csc −1)(csc +1) =1− sin . Then verify the identity.
Because the left side is more complicated, start with it. Begin by multiplying (csc x − 1) by (csc x + 1), and then search for a fundamental identity that can be used to replace the result.
D [-1,1] R [0,π]
y=arccosx
infinite sequence
a function whose domain is the set of positive integers. The function values a1, a2, a3, a4, . . . , an, . . .
sinx=
cos(∏/2 -x)
cot²x + 1=
csc²x
Quotient Identities
tan x = sin x / cos x cot x = cos x / sin x
Explain how to use the fundamental trigonometric identities to find the value of tan u given that sec u = 2 .
Use the Pythagorean identity 1 + tan2u = sec2u. Substitute 2 for the value of sec u and solve for tan u.
Initial Term
It is always assumed to be a1 , unless the problem specifically states otherwise
You cant divide by trig identities and get rid of them because
then you get rid of possible solutions
Graph of sec(t)
..
The nth partial sum of an arithmetic sequence with initial term a1 and common difference d is given by:
Sn = n/2(a1 + an)
tanx=
cot(∏/2 -x)
To prove Trig Identities you need to...
Make sides match by taking one side and simplifying till you get the other side. parts of work along right side of eq sign straight down.
In general, a recursive definition for an arithmetic sequence that begins with a1 may be given by:
a1 given an +1 = an + d (n ≥ 1; "k is an integer" is implied) (n+1 & n are subscripts)
As a special case, zero factorial is defined as
0! = 1
Graph of cos(t)
tan²x + 1=
sec²x
D (-∞,-1]U[1,∞) R [-π/2,0)U(0,π/2]
y=arccscx
D (-∞,-1]U[1,∞) R [0,π/2)U(π/2/π]
y=arcsecx
Guidlines for Verifying Trigonometric Identities
1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends provide insight.
The preliminary goal in solving trigonometric equations is
to isolate the trigonometric function involved in the equation
To solve a trigonometric equation of quadratic type
factor the quadratic, or if this is not possible, use the Quadratic Formula.
To find the first three terms of a sequence, given an expression for its nth term,
evaluate the expression for the nth term at n = 1 to find the first term, at n = 2 for the second term, and so on.
Graph of cot(t)
...
How many solutions does the equation sec x = 2 have? Explain.
The equation has an infinite number of solutions because the secant function has a period of 2π. Any angles coterminal with the equation's solutions on [0, 2π) will also be solutions of the equation. however for tan it is [0,π)
D [-1,1] R [-π/2,π/2]
y=arcsinx
Common Difference
This is denoted by d . It is the number that is always added to a previous term to obtain the following term.
An arithmetic sequence is determined by:
its initial term and its common difference
Pythagorean Identities
sin²x + cos²x = 1 1 + tan²x = sec²x 1 + cot²x = csc²x
Explain how to use the fundamental trigonometric identities to simplify sec x − tan x sin x .
Rewrite the expression in terms of sines and cosines. Combine the resulting fractions to obtain (1 − sin2 x)/(cos x). Using the Pythagorean identity sin2 u + cos2 u = 1, replace the numerator with cos2 x. Simplify the result to obtain cos x.
To solve a trigonometric equation(5.3)
use standard algebraic techniques such as collecting like terms and factoring.
Reciprocal Identities
sin x = 1 / csc x cos x = 1 / sec x tan x = 1 / cot x csc x = 1 / sin x sec x = 1 / cos x cot x = 1 / tan x
Geometric Sequence
A sequence in which each term is found by multiplying the previous term by the same number. Each pair of consecutive numbers have the same ratio
The Fundamental Identities
There are four groups of fundamental identities: reciprocal identities, quotient identities, Pythagorean identities, and even-odd identities. The fundamental identities are used to establish other relationships among trigonometric functions.
The general nth term of an arithmetic sequence with initial term a1 and common difference d is given by:
an = a1 + (n − 1) d Tip: Graph is Linear
If n is a positive integer, n factorial is defined as
n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n − 1) ⋅ n
Section 5
(50 cards)
Foci
fixed points at the centers of an ellipse
Horizontal Shift
(x+5) counter intuitive left 5
Equation of a Circle
(x-h)^2 + (y-k)^2 = r^2
Horizontal Stretch
between 0-1 (.5x) bigger, looks like vertical shrink
Midpoint Formula
Completing the Square Formula
FORMULA FOR THE GENERAL nth TERM OF A GEOMETRIC SEQUENCE
an = a1 ⋅r^(n−1) begin with a1 and keep multiplying by r until we obtain an expression for an Tip: Graph is Exponential
Vertex Formula
Even Function
f(-x)=f(x) Y-Axis Symmetry -x^2= x^2
Distance Formula
Linear Equation (Point-Slope)
Inverse Function
f(f-1(x))=x f-1(f(x))=x
Cases where infinite series dont have a sum
The geometric series 2 + 6 + 18 + 54 + ... has no sum, because: n→∞ Sn = ∞ The geometric series 1− 1+ 1− 1+ ... has no sum, because the partial sums do not approach a single real number.
Vertical Stretch
>1 3(x) bigger, looks like horizontal shrink
Horizontal ellipse
This is the case when the bigger number, a², is under the "x²" term.
An infinite series converges (i.e., has a sum)
The Sn partial sums approach a real number (as n → ∞), which is then called the sum of the series if lim n→∞ Sn = S , where S is a real number, then S is the sum of the series
Quadratic Formula
Standard Equation
c²= a² - b² Where c = foci Where a = Vertices Where b = co vertices
Remember that Sn for a sequence starting with a1 is given by:
Sn = n ∑ak = a1 + a2 +...+ an k=1
Quadratic Equation
Indirect Variation
y=k/x
Can arithmetic sequences converge?
No infinite arithmetic sequence (such as 2 + 5 + 8 + 11+...) can have a sum, unless you include 0 + 0 + 0 + ... as an arithmetic sequence.
Function
1. A relationship from one set (called the domain, x) to another set (called the range, y) that assigns to each element of the domain exactly one element of the range.
Continuous compounding
Increasing the number of compoundings in the compound interest formula without bound leads to continuous compounding, which is given by the formula A = Pert
The nth partial sum of a geometric sequence with initial term a1 and common ratio r (where r ≠ 1) is given by
Sn= a1(1-r^n/1-r) Also could be written with a1 distributed to numerator
Horizontal Shrink
>1 (3x) smaller, looks like vertical stretch
zeros
x-ints
Vertical Shrink
between 0-1 .5(x) smaller, looks like horizontal stretch
How does an infinite geometric series converge?
⇔ (−1 < r < 1) i.e., r <1
x-axis reflection
(x, y) -> (x, -y)
Complex Conjugates
Two complex numbers of the form a + bi and a - bi
Ellipse
Set of points in a plane such that the sum of distances from any point P on the ellipse to two fixed points F1 and F2 (called the foci) is the constant sum, distance = PF1 + PF2 Where, PF1 = distance between a point and F1 and PF2 = distance between the same point and F2
Circle Equation
When you approach a question like, "Given two terms in a geometric sequence find the 8th term and the recursive formula." What do you do?
Divide bigger An by the smaller An and then root the the quotient to the root of bigger An- smaller An to find r. You can then use r to work your way back to A1 and set up the explicit formula
Vertical ellipse
The is the case when the bigger number, a², is under the "y²" term.
Distance formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Odd Function
-(f(x)) Origin Symmetry -(x^3-5x)
Ellipse translation
If h is positive, shift ellipse to the right for horizontal ellipses and up for vertical ellipses. If k is positive, shift ellipse up for horizontal ellipses and to the right for vertical ellipses. Foci and co-vertices will also be translated. Horizontal foci will be h+c and h-c on the x axis and k on the y axis Co-vertices will be k+b, k-b on the y axis and h on the x axis.
After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the formulas:
For n compoundings per year: A = P(1 + r/n)^nt For continuous compounding: A = Pert
The sum of a convergent infinite geometric series with initial term a1 and common ratio r , where −1 < r < 1, lrl<1 is given by:
S = a1/1− r
Vertical Shift
(x^2)+5 up 5
y-axis reflection
(x, y) -> (-x, y)
Linear Equation (Standard)
Major axis
The longer axis of an ellipse that passes through both foci.
In general, a recursive definition for a geometric sequence that begins with a1 may be given by:
a1 given an= an-1 ⋅r (n ≥ 1; "n is an integer" is implied) We assume a1 ≠ 0 and r ≠ 0 .
Linear Equation (Slope-Intercept)
Piecewise- Defined Functions
2 or more ineqaluities for different intervals of the domain
Natural base e
The irrational number e ≈ 2.718281828
Direct Variation
y=kx
Minor axis
The shorter axis of an ellipse that passes through both co-vertices.
Section 6
(50 cards)
Double root
Found when there is only 1 x-int to be solved for.
f(x)=4x^2-5x+3 What is the linear term?
-5x
What are the steps to convert a standard(general) form equation into a turning point? Vice versa?
1.) Isolate the constant. 2.) Make sure the coefficient on the quadratic term is 1. If not, factor out "a". 3.) Create a new constant (by adding 1/2 of the linear coefficient and squaring it on the inside of parentheses, and subtracting it from the constant.) Distribute and add to get to standard form. (Complete the square essentially) (b/2)^2
Difference Quotient
f(x + h) - f(x) / h , h does not equal 0 apply DQ to a given f(x)
To find an inverse algebraically...
1.) Switch variables y and x 2.) isolate y 3.) Label your answer as y^-1 or f^-1.
Horizontal Line Test tells...
(If intersects @ only one point) the original functions has an inverse that is a function.
f(x)=4x^2-5x+3 What is the quadratic term?
4x^2
Domain of Combinations of Functions
Find domain of f(x) & g(x), then the domain of the combination is where they overlap
(f/g)(x)
Divide the equations of f(x) and g(x)
Vertical Reflection
-(x) dont distribute outside of equation reflection over x axis
Discriminant
b^2 -4ac
Function notation
f(x) = output Y
y - y1 = m(x - x1)
point-slope form, for equations
One-to-one correspondence
Passes both the vertical and horizontal tests (graphically).
What are the steps to finding zeros?
1. Set the equation equal to 0. 2. Factor. 3. Set each factor = to 0. 4. Solve for x.
Direction depends on...
whether the coefficient on the x- value (in std. form) is negative or not.
Inverse function definition
a function obtained by expressing the dependent variable of one function as the independent variable of another; f and g are inverse functions if f(x)=y and g(y)=x
Vertex formula
(-b/2a, f(-b/2a))
Piecewise Functions
f(x) = {x + 1, x < 3 ...etc { usually 2. first is the rule, second is constraints on the rule graphically
Odd function
f(-x) = - f(x) , symmetric with origin
f(x)=4x^2-5x+3 What is the constant?
3
Finding the domains of quotients of functions
-find the domains of the 2 given equations separately -combine the two domains to fit the equation you are asked to find (example (f-g)(x)) but make sure that the new domain is not conflicting with the two original domains. -if dividing make sure that the answer will not produce a 0 or a square root of a negative number
y = mx + b
slope-intercept form of a line in a plane, for graphing
Parallel lines
have the same slope
Quadratic Formula
x=(-b±√b²-4ac) /2a Used with std. form.
Relation
a rule of correspondence or pairing of one group to a smaller group
Test for Inverses
The Horizontal Line Test
How to verify if two functions are inverse functions
Plug the equation given into the other equation and get ''x'' as the simplified answer
Horizontal Reflection
2(-(x-2)) distribute negative to entire inside 2(-x+2) reflection over y axis
(f-g)(x)
Subtract the equations f(x) and g(x)
Ax + By = C
standard form, for finding intercepts
Turning Point Form (the different thing from 2.1)
y=a(x-h)^2 +k
(f+g)(x)
Add the equations of f(x) and g(x)
Axis of Symmetry (Turning Pt. Form)
x=h
One to one function
F is one to one if the y value corresponds to exactly one x value. F has an inverse function if and only if it is one to one. (a horizontal line can not intersect the graph of F at more than one point)
To find an inverse graphically...
Make a T-table of the original then make another, switching places with the x- and y- values.
Graph of an inverse function
The two lines when drawn on a coordinate plane look like reflection over y=x
Vertical Line Test
test if a relation is function or not, if it is a function, it only intersects line once
Axis of Symmetry formula
x= -b/2a
(fg)(x)
multiply the equations f(x) and g(x)
Even function
f(-x) = f(x) , symmetric with y-axis
How do you find the vertex... -in standard form? -in turning point form?
-In standard: ( (-b)/(2a), f((-b)/(2a) ); Find the axis of symmetry and plug the value of x into the original equation to find the y value of the vertex. -In turning-point: (h,k)
What's the slope of a line?
rise/run, y2 - y1 / x2 - x1, Δy/Δx
How do you find the x-intercept(s)?
Either: -Quadratic formula, or, -Make y equal 0.
Perpendicular lines
have negative reciprocals m1 = -(1 / m2)
Domain of Composition of Functions
Find the domain of the inside function and the domain of the composition and see where they overlap.
Composition of 2 functions
Written as (f°g)(x) or f(g(x)) -replace the x's of f(x) with the equation g(x) and vise versa
Intercepts
To find the x-intercepts (zeroes): y=0, solve for x. To find y-intercepts: x=0, solve for y.
Standard/ General Form
f(x)= ax^2+bx+c
Vertex (Turning Pt. Form)
(h, k)
Section 7
(50 cards)
rigid transformation
horizontal shifts, vertical shifts and reflections are called __________.
implied domain
If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the __________.
greatest integer
The function f(x)=[[x]] is called the _____ function and is an example of a step function.
How do you determine the direction of a function? (Which way it "opens"?)
Whether "a" is positive of negative; Up if positive, down if negative.
piecewise-defined
the function f(x) = x²-4, x≤0 2x+1, x>0 is an example of a _________ function.
horizontal
A graphical test for the existence of an inverse function is called the _________ line test.
How to add and subtract complex numbers. (2-3i) - (4+6i)
Combine like terms.
How to multiply complex numbers, for example (2+i)(2i-3)
Foil, multiply everything.
c>1, 0<c<1
A nonrigid transformation of y=f(x) represented by cf(x) is a vertical stretch if ________ and a vertical shrink if ________.
zero has an even multiplicity
graph does not cross x axis
At most how many zeros does a function have? Relative extrema?
There are as many zeros as the leading coefficient states; this minus one is the # of relative extrema. n-1 x^5+3 has 4
decreasing
A function f is _________ on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies f(x₁)>f(x₂).
How to solve (2i+3)/(9-i)
Multiply the number and denominator by the conjugate of 9-i witch is 9+i.
zero has an odd multiplicity
graph crosses x axis
Complex conjugate.
The product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form a+bi and a-bi.
pole
the center of the polar graph
What are the simplest polynomials?
Parent functions
quadratic function
The graph of a ____________ is U-shaped.
Solving Inequalities w/ Abs. Value
lx-5l<2 give the constant or "a" the abs value ">-a" & "<a" -2<x-5<2 3<x<7
Solving Double Inequalities
1. Separate into two equations -3<6x-1<3 -3<6x-1 6x-1<3 2. Solve -1/3<x x<2/3 3. Then combine -1/3<x<2/3
What are the traits of graphs of polynomial functions?
Smooth, rounded curves with no sharp turns; Continuous, with no holes, breaks, or gaps.
Repeated Zeros?
Two of the same zeros.
-i
i^3
"r" is radius
the distance from the origin to a point
Consider: f(x)=x^5-3x^4+2x^2+4 - What is the name of this type of polynomial function? - How many real zeros does it have (at most)? - How many relative extrema does it have(at most)?
- Quintic (x^n) - Five (n) - 4 (n-1)
How to find i to any power.
Any number squared is equal to a positive i's value is equal to the value of the square root of -i. Any multiple of the four equals one.
inverse, f-inverse
If the composite functions f(g(x))=x and g(f(x))=x, then the function g is the ______ function of f, and is denoted by _________.
If given the roots, what are the steps to writing a quadratic equation?
1.) Write the root(s) as x=(root) 2.) Take this back to x+/- root=0 3.) Make the results individual factors (and set =0 or f(x)) 4.) Multiply the factors.
Polar Form (Coordinates)
(r, θ)
y=x
the graphs of f and f⁻¹ are reflections of each other in the line _______.
1
i^4
fitting a line to data
The process of finding a linear model for a set of data is called a ___________.
Y-intercept (Higher Degree Polynomials)
put zero in for all X's and solve
-1
i^2
-f(x), f(-x)
A reflection in the x-axis of y=f(x) is represented by h(x)=________, while a reflection in the y-axis is represented by h(x)=________.
Given the vertex and a point on the graph, how do you write a quadratic equation?
From the vertex, x=h and y=k. "Plug in" these into a turning-point form equation. Let the coordinates from the point equal x and y in the equation. Solve for "a", then write an equation using only a, h, and k numerical values.
X-intercept (Higher Degree Polynomials)
when y equals zero (just top)
Leading coefficient test?
Positive Even- Rise left and right Negative Even- Fall left and right Positive Odd- Fall left, rise right Negative Odd- Rise left, fall right
Solving Linear Inequalities
5x-7>3x+9 treat ">" like "=" dividing by a negative will flip the sign ">" to "<"
i
i^1
parallel
Two lines are _________ if and only if their slopes are equal.
range, domain
The domain of f is the _______ of f⁻¹, and the ______ of f⁻¹ is the range of f.
one-to-one
To have an inverse function, a function f must be ________; that is a f(a) = f(b) implies a = b.
V.A. (Higher Degree Polynomials)
when the bottom is zero
1
i^0
independent, dependent
For an equation that represents y as a function of x, the _______ variable is the set of all x in the domain, and the _______ variable is the set of all y in the range.
Can zeros be imaginary/ have imaginary numbers in them?
YES
absolute-value function
The graph of an ____________ is V-shaped.
even
A function is _________ if, for each x in the domain of f, f(-x)=f(x).
minimum
A function value f(a) is a relative ________ of f if there exists an interval (x₁,x₂), containing a such x₁<x<x₂ implies f(a)≤f(x)
Section 8
(50 cards)
Multiplication Principle
Let S be a set of ordered pairs (a, b) of objects, where the first object a comes from a set of size p, and for each choice of object a there are q choices for object b. Then the size of S is p*q
Resultant vector =
The sum of corresponding coordinates from the terminal points of two vectors.
Rectangular Form (Coordinates)
(x, y)
exceptions to row echelon
0 0 0 | 0 in any row means INFINITE SOLUTIONS --------------------------------------------------------------------- 0 0 0 | # in any row means NO SOLUTIONS
Angle theta of a vector =
tangent theta (b / a)
subtract corresponding elements
to subtract two matrices of the same order, ________________
Reference angle theta of a vector =
arctangent (b /a )
θ is Direction Angle
the angle measured counterclockwise from the polar axis
Subtraction Principle
Let U be a larger set containing A. Then, let A^C be the complement of A in U. Then, the number |A| of objects in A is given by |A| = |U| - |A^C|
Vertical component of a vector =
b = | v| sin theta
Horizontal component of a vector =
a = | v | cos theta
elements
the values in the matrix are called _______________
Permutations
-Order matters - nPr ~n = number of total items ~r = number of items being ordered
dividing matrices
cannot divide matrices if you want to move a matrix to the other side of an equation take the inverse of it
Horizontal / Vertical unit vectors =
i = (1, 0), j = (0,1). Any arbitrary vector can be written in terms of v = ai + bj
Addition Principle
Suppose a set S is partition into pairwise disjoint parts Then, the total number of objects in S, |S| is the sum of |S_1| + |S_2| + ... + |S_m|
Finding one unit length vector having same direction as given vector =
Divide components of vector by its magnitude. Result is vector length 1.
Division Principle (pigeon hole principle)
Let S be a finite set that is partitioned into k parts in such a way that each part contains the same number of objects. The number of parts in the partition is given by the rule: k = |S| / (number of objects in a part) ex) There are 740 pigeons in a collection of pigeonholes. If each hole contains 5 pigeons, the number of pigeonholes equals 148.
Counting Principle
(Number of ways of item #1) x (Number of ways of item #2)
n
stands for the amount of columns in a matrix
To change from POLAR to RECTANGULAR coordinates, you use...
x=r cos θ, y= r sin θ
goal of row echelon
work from top to bottom in column and left to right through columns
Polar Curves: Spiral
r = aθ
Operations on vectors =
u + v = (a + c, b + d). Subtraction can be considered adding the negative vector terminal point.
Magnitude of a vector =
| v | = square root of a squared + b squared
multiplying matrices
the inner values have to be the same, if not it is undefined the outer values are the new dimensions of the matrix
add corresponding elements
to add two matrices of the same order, ________________
Polar Curves: Circle - centered at pole
r = a
parameter
a dummy variable, usually t, that is the independent variable in parametric equations x=f(t) and y=f(t).
scalar
referring to the number that a matrix is multiplied by
equal
when two matrices have the same order and corresponding elements are equal then the matrix is ______________
scalar multiplication
when you times each element in a matrix by the same value, it is called _____________________
If no angle of θ is determined...
the graph is a circle. The "r" value found is the radius of the circle.
multiplied
when the columns of the first matrix is equal to the rows of the second matrix, the matrices can be ____________________
zero matrix
a matrix which has all elements zero
parametric equations
any set of equations using a parameter typically used to describe motion such as x=f(t) and y=f(t)
RC COLA !!
row x column
inverse equation of a 2 x 2 matrix
If only ONE variable in rectangular is found x or y...
x = a is a vertical line y = a is a horizontal line
Polar Curves: Circle - shifted along the y-axis
r = a sin θ
rules of row echelon
1. switch any two rows - (R1 <------> R2) 2. add any two rows (! cannot subtract !) - (R1 + R2 -----> R1 or R2) 3. multiply any row by a non-zero number - (#R1-----> R1) 4. combine steps 2 and 3 into one move - (#R1 + R2 -----> R1 or R2)
m x n
the order of the matrix
Eliminating the parameter
the process of solving one parametric equation for t so you may substitute that equation into the other parametric equation for t to create a rectangular equation where y=f(x) Example: if x=t-4 and y=¼t solve the first equation for t so t=x-4 then by substitution y=¼(x-4) or y=¼x-1.
NO
Is matrix multiplication commutative?
matrix
a rectangular array of numbers arranged in rows and columns
adding and subtracting matrices
have to have the same dimensions
To change from RECTANGULAR to POLAR, you use...
r² = x² + y² tan θ =|y|/|x| ; (x≠0) remember tangent equation gives you reference angle only. Must adjust if not in Quadrant I
Polar Curves: Circle - shifted along the x-axis
r = a cos θ
detailed matrix form
labeling the rows and columns of a matrix
m
stands for the amount of rows in a matrix
Section 9
(50 cards)
x²=4py (parabola)
+ opens up - opens down
ellipse
the set of all points P such that the sum of the distances between P and two distinct fixed points (foci) is a constant. - vertex is always bigger than covertex - vertex and focus are always found on major axis
co-vertices for horizontal ellipse
(h,k±b)
Formula for up/down parabola
y=a(x-h)²+k
Combinations
-Order does not matter -nCr ~n = number of total items ~r = number of items being combined
Independent Probability
Probability of A and B P(A) x P(B)
What happens as e approaches infinity?
The conic becomes less circular and more stretched out
formula for minor axis for both
2b
What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC < 0?
Hyperbola
Formula for directrix of left/right parabola
x=h−1/4a
focus for horizontal ellipse
(h±c,k)
Axis of symmetry formula for up/down parabola
x=h
Centers for both
(h,k)
formula for major axis for both
2a
Formula for focus of Up/Down Parabola
(h,k+1/4a)
x²+y²=r² (circle)
the set of all points that are equidistant from a fixed point called the center. center (0,0) radius r
What is the conic with an e < 1?
Ellipse
vertices for left/right hyperbola
(h±a,k)
vertices for vertical ellipse
(h,k±a)
Formula for left/right parabola
x=a(y-k)²+h
Formula for directrix of up/down parabola
y=k−1/4a
formula for hyperbola opens up/down
What happens as e approaches 0?
The conic becomes more and more like a circle
What is the conic with an e > 1?
Hyperbola
Center of circle
(h,k)
How to determine direction of left/right parabola
If a>0, opens right. If a<0, opens left
Radius
r
y²=4px (parabola)
+ opens right - opens left
parabola
the set of all points that are equidistant from the focus and directrix. p= distance from vertex to focus and from vertex to directrix.
vertices for up/down hyperbola
(h,k±a)
Formula for focus of left/right parabola
(h+1/4a,k)
Formula for circle
(x-h)²+(y-k)² = r²
Polar equation of a conic
r = ep / 1 - e cos(θ)
co-vertices for vertical ellipse
(h±b,k)
Definition of eccentricity
c/a
What is the conic with an e = 1?
Parabola
slope for asymptote for up/down
±a/b
formula for horizontal ellipse
Dependent Probability
Probability of A or B P(A) + P(B) - Overlap (Possibilities that fall into A and B)
vertices for horizontal ellipse
(h±a,k)
What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC = 0?
Parabola
formula for hyperbola opens left/right
What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC > 0?
Ellipse (or circle)
How to determine direction of up/down parabola
If a>0, opens up. If a<0 opens down.
slope of asymptote for left/right hyperbola
±b/a
formula for vertical ellipse
focus for vertical ellipse
(h,k±c)
hyperbola
the set of all points P such that the difference of the distances between P and two distinct fixed points (foci) is a constant. - vertex is always in front
Axis of symmetry formula for left/right parabola
y=k
Permutation Formula
You have n items and want to find the number of ways r items can be ordered
Section 10
(10 cards)