Section 1

Preview this deck

Cot=

Front

Star 0%
Star 0%
Star 0%
Star 0%
Star 0%

0.0

0 reviews

5
0
4
0
3
0
2
0
1
0

Active users

0

All-time users

0

Favorites

0

Last updated

1 year ago

Date created

Mar 1, 2020

Cards (460)

Section 1

(50 cards)

Cot=

Front

X/y reciprocal of tan y/x

Back

The Change-of-Base Property: Introducing Common and Natural Logarithms

Front

Introducing Common Logarithms Introducing Natural Logarithms

Back

The Product Rule

Front

Let b, M, and N be positive real numbers with b 1. The logarithm of a product is the sum of the logarithms.

Back

Tangent

Front

O/A or Y/x

Back

60°

Front

Pi/3

Back

sin ø=?

Front

cos (90-ø)

Back

Condensing Logarithmic Expressions

Front

Back

sin 72

Front

cos 18

Back

Sine=

Front

O/H, or y

Back

0° or 360°

Front

2pi

Back

30°

Front

Pi/6

Back

The Power Rule

Front

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.

Back

If terminal rotates clockwise

Front

Produces a negative angle

Back

Complimentary angles

Front

Add up to 90° if it is greater than pi/2, it has no compliment

Back

The Quotient Rule

Front

Let b, M, and N be positive real numbers with b 1. The logarithm of a quotient is the difference of the logarithms.

Back

Tan and cot are (even or odd?)

Front

Odd

Back

90°

Front

Pi/2

Back

Periodic function

Front

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)

Back

cos ø=?

Front

sin (90-ø)

Back

Circumference

Front

2pir

Back

The domain of cos

Front

All real numbers

Back

Radian

Front

Used as a new way to measure angles in terms of pi.

Back

Properties of Common Logarithms

Front

Back

Inverse Properties of Logarithms

Front

For b > 0 and b 1, log base b, b∧x = x b∧log bx = x

Back

The Domain of a Logarithmic Function

Front

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.

Back

Degree to radian

Front

Degree • pi/180

Back

Cos (even or odd?)

Front

Even

Back

Coterminal angles

Front

Angles in the same position

Back

Csc=

Front

1/y or the reciprocal of y(sin)

Back

The domain of sin

Front

All real numbers

Back

Sec=

Front

1/x or the reciprocal of x (cos)

Back

Sine (even or odd?)

Front

Odd

Back

Natural Logarithms

Front

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

Back

sin 10

Front

cos 80

Back

Characteristics of Logarithmic functions of the Form f(x) = log base b, x

Front

Back

Common Logarithms

Front

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

Back

Range of cos

Front

Between -1>x< 1 shit be happening "equal to -1 & 1 as well"

Back

cos 70

Front

sin 20

Back

Properties of Natural Logarithms

Front

Back

Radian to degree

Front

Radian measurement • 180/pi

Back

Half rotation

Front

180° or pi

Back

45°

Front

Pi/4

Back

Definition of the Logarithmic Function

Front

Definition of the Logarithmic Function

Back

The Change-of-Base Property

Front

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.

Back

If terminal rotates counter clockwise

Front

Produces a positive angle

Back

Cosine=

Front

A/H, or x

Back

Range of sin

Front

-1>y<1 is where the shit goes down "less than or equal to"

Back

Basic Logarithmic Properties Involving One

Front

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)

Back

Full rotation

Front

360° or 2pi

Back

Supplementary angles

Front

Adds to 180°. If an angle is greater than pi, it has no supplement

Back

Section 2

(50 cards)

special right triangles

Front

there are two special right triangles: 30-60-90 and 45-45-90

Back

Phase Shift

Front

bx+/-c=0 bx+/-c=2π

Back

cosθ=

Front

x/r

Back

Cos (and sec) is positive in quadrants?

Front

I and IV

Back

Domain

Front

ℝ (-∞, ∞)

Back

a > 1

Front

vertical stretch

Back

Area of a Sector

Front

A=1/2(r^2)θ θ must be in radians

Back

0⁰

Front

0 (1,0)

Back

Even Functions When Neg.

Front

cos(-θ)=cosθ sec(-θ)=secθ they're the same as the non-negated outcome so you should treat them as such

Back

b is negative

Front

vertical flip across y-axis (in cosine, no visual change)

Back

r=

Front

√(x²+y²)

Back

b < 1

Front

less cycles

Back

y = A sin(Bx) + C

Front

A: amplitude (vertical stretch/compression C: midline (vertical shift) B= 2π/ period (horizontal stretch/compression)

Back

horizontal shift

Front

(x = h)

Back

cotθ=

Front

x/y, y≠0

Back

Period

Front

2π/|b|

Back

Reference angle θ'

Front

The acute angle formed by the terminal side of the angle and the horizontal axis.

Back

Tan (and cot) is positive in quadrants?

Front

I and III

Back

Cosine Curve

Front

Starts at amplitude

Back

unit circle

Front

Back

k is positve

Front

shift up

Back

60⁰

Front

π/3 (1/2, √3/2)

Back

Arc Length

Front

S=rθ θ must be in radians

Back

Sin (and csc) is positive in quadrants?

Front

I and II

Back

a < 1

Front

vertical shrink

Back

Unit Circle

Front

x^2+y^2=1 points on circle must satisfy equation

Back

secθ=

Front

r/x, x≠0

Back

If θ is in Quadrant I, then θ' is

Front

θ'=θ

Back

a is negative

Front

horizontal flip on x-axis

Back

45⁰

Front

π/4 (√2/2, √2/2)

Back

tanθ=

Front

y/x, x≠0

Back

90⁰

Front

π/2 (0,1)

Back

Sine Curve

Front

Starts at 0

Back

increments

Front

period / 4

Back

graph cos(x)

Front

Back

k is negative

Front

shift down

Back

cscθ=

Front

r/y, y≠0

Back

Amplitude (in terms of max & min)

Front

(max + min) / 2

Back

vertical shift

Front

(y = k)

Back

30⁰

Front

π/6 (√3/2, 1/2)

Back

reference angles

Front

the acute angle formed by the terminal side of the angle and the x-axis

Back

sinθ=

Front

y/r

Back

Range

Front

[-a, a]

Back

Odd Functions

Front

sin(-θ)= -(sinθ) csc(-θ)= -(cscθ) tan(-θ)= -(tanθ) cot(-θ)= -(cotθ) the same as the non-negated but just negative.

Back

b > 1

Front

more cycles

Back

reference right triangles

Front

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

Back

b

Front

period

Back

Increments

Front

period/4

Back

graph sin(x)

Front

Back

a

Front

amplitude

Back

Section 3

(50 cards)

arccos

Front

Domain: [-1, 1] Range: [0, π]

Back

Period of csc(t)

Front

Period: 2π (same as sin(t))

Back

- cos x

Front

Back

Domain of cot(t)

Front

Domain: All real numbers other than nπ for any integer, n. (same as csc(t))

Back

Solving a Right Triangle

Front

Finding all side lengths and angle measures of a triangle.

Back

phase shift =

Front

c/|b|

Back

2 cos x

Front

Back

- cos(x) + 1

Front

Back

Equation of csc(t)

Front

t=1/y

Back

sin x

Front

Back

Period of sin(t)

Front

Period: 2π (same as cos(t))

Back

Period of cot(t)

Front

Period: 2π (same as csc(t))

Back

Range of sin(t)

Front

Range: [−1,1] (same as cos(t))

Back

Equation of sec(t)

Front

t=1/x

Back

Domain of tan(t)

Front

Domain: All real numbers other than π/2+nπ for any integer n. (same as sec(t))

Back

- sin x

Front

Back

Domain of sin(t)

Front

Domain: All reals (−∞, ∞) (same as cos(t))

Back

Range of tan(t)

Front

Range: All reals (−∞, ∞) (same as sec(t))

Back

arcsin

Front

Domain: [-1, 1] Range: [-π/2, π/2]

Back

If θ is in Quadrant IV, then θ' is

Front

θ'=360⁰- θ θ'= 2π - θ

Back

Range of cos(t)

Front

Range: [−1,1] (same as sin(t))

Back

Range of sec(t)

Front

Range: All reals (−∞, ∞) (same as tan(t))

Back

Equation of cot(t)

Front

t=x/y

Back

Range of csc(t)

Front

Range: (-∞,-1]∪[1,∞) (same as cot(t))

Back

cos x

Front

Back

arctan

Front

Domain: [-∞, ∞] Range: [-π/2, π/2]

Back

Graph of csc(t)

Front

...

Back

- sin(x) + 2

Front

Back

Inverse Trigonometric Ratios

Front

You can use sin⁻¹, cos⁻¹ and tan⁻¹ to find the measure of an angle in a right triangle given its sides.

Back

If θ is in Quadrant II, then θ' is

Front

θ'=180⁰- θ θ'= π - θ

Back

arccot x

Front

D: (-∞,∞) R: (0,π)

Back

Period of tan(t)

Front

Period: π (same as cot(t))

Back

Equation of cos(t)

Front

t=x

Back

sin(2x)

Front

Back

Arctan(1)

Front

tan(x)=1 x=π/4

Back

Graph of sin(t)

Front

Back

Trigonometric Ratio

Front

A ratio of the lengths of two sides in a right triangle to the angle.

Back

Range of cot(t)

Front

Range: (-∞,-1]∪[1,∞) (same as csc(t))

Back

Equation of tan(t)

Front

t=y/x

Back

arccsc x

Front

D: (⁻∞ , -1] U [1 , + ∞) R: (-π/2, 0) U (0, π/2]

Back

Domain of csc(t)

Front

Domain: All real numbers other than nπ for any integer, n. (same as cot(t))

Back

Domain of sec(t)

Front

Domain: All real numbers other than π/2+nπ for any integer n. (same as tan(t))

Back

Domain of cos(t)

Front

Domain: All reals (−∞, ∞) (same as sin(t))

Back

Steps for Csc(x) and Sec(x) Graphs

Front

1. Graph Reciprocal 2. Everywhere you see x-int draw asymptotes. 3. Draw parabolas on the maxs and mins.

Back

arcsec x

Front

D: (⁻∞ , -1] U [1 , + ∞) R: [0 , π/2) U (π/2 , π]

Back

Compositon of Functions

Front

Back

Graph of tan(t)

Front

...

Back

Period of cos(t)

Front

Period: 2π (same as sin(t))

Back

Equation of sin(t)

Front

t=y

Back

If θ is in Quadrant III, then θ' is

Front

θ'= θ - 180⁰ θ'= θ - π

Back

Section 4

(50 cards)

cosx=

Front

sin(∏/2 -x)

Back

Care must be taken when squaring both sides of a trigonometric equation to obtain a quadratic because

Front

this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous

Back

finite sequence

Front

a sequence whose domain consists of the first n positive integers only

Back

D (-∞,∞) R (-π/2,π/2)

Front

y=arctanx

Back

The key to verifying identities is

Front

the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions.

Back

D (-∞,∞) R (0,π)

Front

y=arccotx

Back

A geometric sequence is determined by:

Front

Its initial term and its common ratio.

Back

sin²x + cos²s=

Front

1

Back

To solve an equation in which two or more trigonometric functions occur

Front

collect all terms on one side and try to separate the functions by factoring or by using appropriate identities.

Back

cscx=

Front

sec(∏/2 -x)

Back

Tricks when proving Trig Identities

Front

Conjugate, factoring, and common denominator

Back

Arithmetic Sequence

Front

A sequence in which each term is found by adding a fixed amount to the previous term

Back

Even-Odd Identities

Front

sin(-x) = - sin x cos(-x) = cos x tan (-x) = - tan x csc (-x) = - csc x sec (-x) = sec x cot (-x) = - cot x

Back

Common Ratio

Front

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

Back

Describe a strategy for verifying the identity 2 2 sin (csc −1)(csc +1) =1− sin . Then verify the identity.

Front

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.

Back

D [-1,1] R [0,π]

Front

y=arccosx

Back

infinite sequence

Front

a function whose domain is the set of positive integers. The function values a1, a2, a3, a4, . . . , an, . . .

Back

sinx=

Front

cos(∏/2 -x)

Back

cot²x + 1=

Front

csc²x

Back

Quotient Identities

Front

tan x = sin x / cos x cot x = cos x / sin x

Back

Explain how to use the fundamental trigonometric identities to find the value of tan u given that sec u = 2 .

Front

Use the Pythagorean identity 1 + tan2u = sec2u. Substitute 2 for the value of sec u and solve for tan u.

Back

Initial Term

Front

It is always assumed to be a1 , unless the problem specifically states otherwise

Back

You cant divide by trig identities and get rid of them because

Front

then you get rid of possible solutions

Back

Graph of sec(t)

Front

..

Back

The nth partial sum of an arithmetic sequence with initial term a1 and common difference d is given by:

Front

Sn = n/2(a1 + an)

Back

tanx=

Front

cot(∏/2 -x)

Back

To prove Trig Identities you need to...

Front

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.

Back

In general, a recursive definition for an arithmetic sequence that begins with a1 may be given by:

Front

a1 given an +1 = an + d (n ≥ 1; "k is an integer" is implied) (n+1 & n are subscripts)

Back

As a special case, zero factorial is defined as

Front

0! = 1

Back

Graph of cos(t)

Front

Back

tan²x + 1=

Front

sec²x

Back

D (-∞,-1]U[1,∞) R [-π/2,0)U(0,π/2]

Front

y=arccscx

Back

D (-∞,-1]U[1,∞) R [0,π/2)U(π/2/π]

Front

y=arcsecx

Back

Guidlines for Verifying Trigonometric Identities

Front

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.

Back

The preliminary goal in solving trigonometric equations is

Front

to isolate the trigonometric function involved in the equation

Back

To solve a trigonometric equation of quadratic type

Front

factor the quadratic, or if this is not possible, use the Quadratic Formula.

Back

To find the first three terms of a sequence, given an expression for its nth term,

Front

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.

Back

Graph of cot(t)

Front

...

Back

How many solutions does the equation sec x = 2 have? Explain.

Front

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)

Back

D [-1,1] R [-π/2,π/2]

Front

y=arcsinx

Back

Common Difference

Front

This is denoted by d . It is the number that is always added to a previous term to obtain the following term.

Back

An arithmetic sequence is determined by:

Front

its initial term and its common difference

Back

Pythagorean Identities

Front

sin²x + cos²x = 1 1 + tan²x = sec²x 1 + cot²x = csc²x

Back

Explain how to use the fundamental trigonometric identities to simplify sec x − tan x sin x .

Front

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.

Back

To solve a trigonometric equation(5.3)

Front

use standard algebraic techniques such as collecting like terms and factoring.

Back

Reciprocal Identities

Front

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

Back

Geometric Sequence

Front

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

Back

The Fundamental Identities

Front

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.

Back

The general nth term of an arithmetic sequence with initial term a1 and common difference d is given by:

Front

an = a1 + (n − 1) d Tip: Graph is Linear

Back

If n is a positive integer, n factorial is defined as

Front

n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n − 1) ⋅ n

Back

Section 5

(50 cards)

Foci

Front

fixed points at the centers of an ellipse

Back

Horizontal Shift

Front

(x+5) counter intuitive left 5

Back

Equation of a Circle

Front

(x-h)^2 + (y-k)^2 = r^2

Back

Horizontal Stretch

Front

between 0-1 (.5x) bigger, looks like vertical shrink

Back

Midpoint Formula

Front

Back

Completing the Square Formula

Front

Back

FORMULA FOR THE GENERAL nth TERM OF A GEOMETRIC SEQUENCE

Front

an = a1 ⋅r^(n−1) begin with a1 and keep multiplying by r until we obtain an expression for an Tip: Graph is Exponential

Back

Vertex Formula

Front

Back

Even Function

Front

f(-x)=f(x) Y-Axis Symmetry -x^2= x^2

Back

Distance Formula

Front

Back

Linear Equation (Point-Slope)

Front

Back

Inverse Function

Front

f(f-1(x))=x f-1(f(x))=x

Back

Cases where infinite series dont have a sum

Front

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.

Back

Vertical Stretch

Front

>1 3(x) bigger, looks like horizontal shrink

Back

Horizontal ellipse

Front

This is the case when the bigger number, a², is under the "x²" term.

Back

An infinite series converges (i.e., has a sum)

Front

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

Back

Quadratic Formula

Front

Back

Standard Equation

Front

c²= a² - b² Where c = foci Where a = Vertices Where b = co vertices

Back

Remember that Sn for a sequence starting with a1 is given by:

Front

Sn = n ∑ak = a1 + a2 +...+ an k=1

Back

Quadratic Equation

Front

Back

Indirect Variation

Front

y=k/x

Back

Can arithmetic sequences converge?

Front

No infinite arithmetic sequence (such as 2 + 5 + 8 + 11+...) can have a sum, unless you include 0 + 0 + 0 + ... as an arithmetic sequence.

Back

Function

Front

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.

Back

Continuous compounding

Front

Increasing the number of compoundings in the compound interest formula without bound leads to continuous compounding, which is given by the formula A = Pert

Back

The nth partial sum of a geometric sequence with initial term a1 and common ratio r (where r ≠ 1) is given by

Front

Sn= a1(1-r^n/1-r) Also could be written with a1 distributed to numerator

Back

Horizontal Shrink

Front

>1 (3x) smaller, looks like vertical stretch

Back

zeros

Front

x-ints

Back

Vertical Shrink

Front

between 0-1 .5(x) smaller, looks like horizontal stretch

Back

How does an infinite geometric series converge?

Front

⇔ (−1 < r < 1) i.e., r <1

Back

x-axis reflection

Front

(x, y) -> (x, -y)

Back

Complex Conjugates

Front

Two complex numbers of the form a + bi and a - bi

Back

Ellipse

Front

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

Back

Circle Equation

Front

Back

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?

Front

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

Back

Vertical ellipse

Front

The is the case when the bigger number, a², is under the "y²" term.

Back

Distance formula

Front

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Back

Odd Function

Front

-(f(x)) Origin Symmetry -(x^3-5x)

Back

Ellipse translation

Front

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.

Back

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:

Front

For n compoundings per year: A = P(1 + r/n)^nt For continuous compounding: A = Pert

Back

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:

Front

S = a1/1− r

Back

Vertical Shift

Front

(x^2)+5 up 5

Back

y-axis reflection

Front

(x, y) -> (-x, y)

Back

Linear Equation (Standard)

Front

Back

Major axis

Front

The longer axis of an ellipse that passes through both foci.

Back

In general, a recursive definition for a geometric sequence that begins with a1 may be given by:

Front

a1 given an= an-1 ⋅r (n ≥ 1; "n is an integer" is implied) We assume a1 ≠ 0 and r ≠ 0 .

Back

Linear Equation (Slope-Intercept)

Front

Back

Piecewise- Defined Functions

Front

2 or more ineqaluities for different intervals of the domain

Back

Natural base e

Front

The irrational number e ≈ 2.718281828

Back

Direct Variation

Front

y=kx

Back

Minor axis

Front

The shorter axis of an ellipse that passes through both co-vertices.

Back

Section 6

(50 cards)

Double root

Front

Found when there is only 1 x-int to be solved for.

Back

f(x)=4x^2-5x+3 What is the linear term?

Front

-5x

Back

What are the steps to convert a standard(general) form equation into a turning point? Vice versa?

Front

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

Back

Difference Quotient

Front

f(x + h) - f(x) / h , h does not equal 0 apply DQ to a given f(x)

Back

To find an inverse algebraically...

Front

1.) Switch variables y and x 2.) isolate y 3.) Label your answer as y^-1 or f^-1.

Back

Horizontal Line Test tells...

Front

(If intersects @ only one point) the original functions has an inverse that is a function.

Back

f(x)=4x^2-5x+3 What is the quadratic term?

Front

4x^2

Back

Domain of Combinations of Functions

Front

Find domain of f(x) & g(x), then the domain of the combination is where they overlap

Back

(f/g)(x)

Front

Divide the equations of f(x) and g(x)

Back

Vertical Reflection

Front

-(x) dont distribute outside of equation reflection over x axis

Back

Discriminant

Front

b^2 -4ac

Back

Function notation

Front

f(x) = output Y

Back

y - y1 = m(x - x1)

Front

point-slope form, for equations

Back

One-to-one correspondence

Front

Passes both the vertical and horizontal tests (graphically).

Back

What are the steps to finding zeros?

Front

1. Set the equation equal to 0. 2. Factor. 3. Set each factor = to 0. 4. Solve for x.

Back

Direction depends on...

Front

whether the coefficient on the x- value (in std. form) is negative or not.

Back

Inverse function definition

Front

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

Back

Vertex formula

Front

(-b/2a, f(-b/2a))

Back

Piecewise Functions

Front

f(x) = {x + 1, x < 3 ...etc { usually 2. first is the rule, second is constraints on the rule graphically

Back

Odd function

Front

f(-x) = - f(x) , symmetric with origin

Back

f(x)=4x^2-5x+3 What is the constant?

Front

3

Back

Finding the domains of quotients of functions

Front

-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

Back

y = mx + b

Front

slope-intercept form of a line in a plane, for graphing

Back

Parallel lines

Front

have the same slope

Back

Quadratic Formula

Front

x=(-b±√b²-4ac) /2a Used with std. form.

Back

Relation

Front

a rule of correspondence or pairing of one group to a smaller group

Back

Test for Inverses

Front

The Horizontal Line Test

Back

How to verify if two functions are inverse functions

Front

Plug the equation given into the other equation and get ''x'' as the simplified answer

Back

Horizontal Reflection

Front

2(-(x-2)) distribute negative to entire inside 2(-x+2) reflection over y axis

Back

(f-g)(x)

Front

Subtract the equations f(x) and g(x)

Back

Ax + By = C

Front

standard form, for finding intercepts

Back

Turning Point Form (the different thing from 2.1)

Front

y=a(x-h)^2 +k

Back

(f+g)(x)

Front

Add the equations of f(x) and g(x)

Back

Axis of Symmetry (Turning Pt. Form)

Front

x=h

Back

One to one function

Front

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)

Back

To find an inverse graphically...

Front

Make a T-table of the original then make another, switching places with the x- and y- values.

Back

Graph of an inverse function

Front

The two lines when drawn on a coordinate plane look like reflection over y=x

Back

Vertical Line Test

Front

test if a relation is function or not, if it is a function, it only intersects line once

Back

Axis of Symmetry formula

Front

x= -b/2a

Back

(fg)(x)

Front

multiply the equations f(x) and g(x)

Back

Even function

Front

f(-x) = f(x) , symmetric with y-axis

Back

How do you find the vertex... -in standard form? -in turning point form?

Front

-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)

Back

What's the slope of a line?

Front

rise/run, y2 - y1 / x2 - x1, Δy/Δx

Back

How do you find the x-intercept(s)?

Front

Either: -Quadratic formula, or, -Make y equal 0.

Back

Perpendicular lines

Front

have negative reciprocals m1 = -(1 / m2)

Back

Domain of Composition of Functions

Front

Find the domain of the inside function and the domain of the composition and see where they overlap.

Back

Composition of 2 functions

Front

Written as (f°g)(x) or f(g(x)) -replace the x's of f(x) with the equation g(x) and vise versa

Back

Intercepts

Front

To find the x-intercepts (zeroes): y=0, solve for x. To find y-intercepts: x=0, solve for y.

Back

Standard/ General Form

Front

f(x)= ax^2+bx+c

Back

Vertex (Turning Pt. Form)

Front

(h, k)

Back

Section 7

(50 cards)

rigid transformation

Front

horizontal shifts, vertical shifts and reflections are called __________.

Back

implied domain

Front

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 __________.

Back

greatest integer

Front

The function f(x)=[[x]] is called the _____ function and is an example of a step function.

Back

How do you determine the direction of a function? (Which way it "opens"?)

Front

Whether "a" is positive of negative; Up if positive, down if negative.

Back

piecewise-defined

Front

the function f(x) = x²-4, x≤0 2x+1, x>0 is an example of a _________ function.

Back

horizontal

Front

A graphical test for the existence of an inverse function is called the _________ line test.

Back

How to add and subtract complex numbers. (2-3i) - (4+6i)

Front

Combine like terms.

Back

How to multiply complex numbers, for example (2+i)(2i-3)

Front

Foil, multiply everything.

Back

c>1, 0<c<1

Front

A nonrigid transformation of y=f(x) represented by cf(x) is a vertical stretch if ________ and a vertical shrink if ________.

Back

zero has an even multiplicity

Front

graph does not cross x axis

Back

At most how many zeros does a function have? Relative extrema?

Front

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

Back

decreasing

Front

A function f is _________ on an interval if, for any x₁ and x₂ in the interval, x₁<x₂ implies f(x₁)>f(x₂).

Back

How to solve (2i+3)/(9-i)

Front

Multiply the number and denominator by the conjugate of 9-i witch is 9+i.

Back

zero has an odd multiplicity

Front

graph crosses x axis

Back

Complex conjugate.

Front

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.

Back

pole

Front

the center of the polar graph

Back

What are the simplest polynomials?

Front

Parent functions

Back

quadratic function

Front

The graph of a ____________ is U-shaped.

Back

Solving Inequalities w/ Abs. Value

Front

lx-5l<2 give the constant or "a" the abs value ">-a" & "<a" -2<x-5<2 3<x<7

Back

Solving Double Inequalities

Front

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

Back

What are the traits of graphs of polynomial functions?

Front

Smooth, rounded curves with no sharp turns; Continuous, with no holes, breaks, or gaps.

Back

Repeated Zeros?

Front

Two of the same zeros.

Back

-i

Front

i^3

Back

"r" is radius

Front

the distance from the origin to a point

Back

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)?

Front

- Quintic (x^n) - Five (n) - 4 (n-1)

Back

How to find i to any power.

Front

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.

Back

inverse, f-inverse

Front

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 _________.

Back

If given the roots, what are the steps to writing a quadratic equation?

Front

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.

Back

Polar Form (Coordinates)

Front

(r, θ)

Back

y=x

Front

the graphs of f and f⁻¹ are reflections of each other in the line _______.

Back

1

Front

i^4

Back

fitting a line to data

Front

The process of finding a linear model for a set of data is called a ___________.

Back

Y-intercept (Higher Degree Polynomials)

Front

put zero in for all X's and solve

Back

-1

Front

i^2

Back

-f(x), f(-x)

Front

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)=________.

Back

Given the vertex and a point on the graph, how do you write a quadratic equation?

Front

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.

Back

X-intercept (Higher Degree Polynomials)

Front

when y equals zero (just top)

Back

Leading coefficient test?

Front

Positive Even- Rise left and right Negative Even- Fall left and right Positive Odd- Fall left, rise right Negative Odd- Rise left, fall right

Back

Solving Linear Inequalities

Front

5x-7>3x+9 treat ">" like "=" dividing by a negative will flip the sign ">" to "<"

Back

i

Front

i^1

Back

parallel

Front

Two lines are _________ if and only if their slopes are equal.

Back

range, domain

Front

The domain of f is the _______ of f⁻¹, and the ______ of f⁻¹ is the range of f.

Back

one-to-one

Front

To have an inverse function, a function f must be ________; that is a f(a) = f(b) implies a = b.

Back

V.A. (Higher Degree Polynomials)

Front

when the bottom is zero

Back

1

Front

i^0

Back

independent, dependent

Front

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.

Back

Can zeros be imaginary/ have imaginary numbers in them?

Front

YES

Back

absolute-value function

Front

The graph of an ____________ is V-shaped.

Back

even

Front

A function is _________ if, for each x in the domain of f, f(-x)=f(x).

Back

minimum

Front

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)

Back

Section 8

(50 cards)

Multiplication Principle

Front

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

Back

Resultant vector =

Front

The sum of corresponding coordinates from the terminal points of two vectors.

Back

Rectangular Form (Coordinates)

Front

(x, y)

Back

exceptions to row echelon

Front

0 0 0 | 0 in any row means INFINITE SOLUTIONS --------------------------------------------------------------------- 0 0 0 | # in any row means NO SOLUTIONS

Back

Angle theta of a vector =

Front

tangent theta (b / a)

Back

subtract corresponding elements

Front

to subtract two matrices of the same order, ________________

Back

Reference angle theta of a vector =

Front

arctangent (b /a )

Back

θ is Direction Angle

Front

the angle measured counterclockwise from the polar axis

Back

Subtraction Principle

Front

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|

Back

Vertical component of a vector =

Front

b = | v| sin theta

Back

Horizontal component of a vector =

Front

a = | v | cos theta

Back

elements

Front

the values in the matrix are called _______________

Back

Permutations

Front

-Order matters - nPr ~n = number of total items ~r = number of items being ordered

Back

dividing matrices

Front

cannot divide matrices if you want to move a matrix to the other side of an equation take the inverse of it

Back

Horizontal / Vertical unit vectors =

Front

i = (1, 0), j = (0,1). Any arbitrary vector can be written in terms of v = ai + bj

Back

Addition Principle

Front

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|

Back

Finding one unit length vector having same direction as given vector =

Front

Divide components of vector by its magnitude. Result is vector length 1.

Back

Division Principle (pigeon hole principle)

Front

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.

Back

Counting Principle

Front

(Number of ways of item #1) x (Number of ways of item #2)

Back

n

Front

stands for the amount of columns in a matrix

Back

To change from POLAR to RECTANGULAR coordinates, you use...

Front

x=r cos θ, y= r sin θ

Back

goal of row echelon

Front

work from top to bottom in column and left to right through columns

Back

Polar Curves: Spiral

Front

r = aθ

Back

Operations on vectors =

Front

u + v = (a + c, b + d). Subtraction can be considered adding the negative vector terminal point.

Back

Magnitude of a vector =

Front

| v | = square root of a squared + b squared

Back

multiplying matrices

Front

the inner values have to be the same, if not it is undefined the outer values are the new dimensions of the matrix

Back

add corresponding elements

Front

to add two matrices of the same order, ________________

Back

Polar Curves: Circle - centered at pole

Front

r = a

Back

parameter

Front

a dummy variable, usually t, that is the independent variable in parametric equations x=f(t) and y=f(t).

Back

scalar

Front

referring to the number that a matrix is multiplied by

Back

equal

Front

when two matrices have the same order and corresponding elements are equal then the matrix is ______________

Back

scalar multiplication

Front

when you times each element in a matrix by the same value, it is called _____________________

Back

If no angle of θ is determined...

Front

the graph is a circle. The "r" value found is the radius of the circle.

Back

multiplied

Front

when the columns of the first matrix is equal to the rows of the second matrix, the matrices can be ____________________

Back

zero matrix

Front

a matrix which has all elements zero

Back

parametric equations

Front

any set of equations using a parameter typically used to describe motion such as x=f(t) and y=f(t)

Back

RC COLA !!

Front

row x column

Back

inverse equation of a 2 x 2 matrix

Front

Back

If only ONE variable in rectangular is found x or y...

Front

x = a is a vertical line y = a is a horizontal line

Back

Polar Curves: Circle - shifted along the y-axis

Front

r = a sin θ

Back

rules of row echelon

Front

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)

Back

m x n

Front

the order of the matrix

Back

Eliminating the parameter

Front

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.

Back

NO

Front

Is matrix multiplication commutative?

Back

matrix

Front

a rectangular array of numbers arranged in rows and columns

Back

adding and subtracting matrices

Front

have to have the same dimensions

Back

To change from RECTANGULAR to POLAR, you use...

Front

r² = x² + y² tan θ =|y|/|x| ; (x≠0) remember tangent equation gives you reference angle only. Must adjust if not in Quadrant I

Back

Polar Curves: Circle - shifted along the x-axis

Front

r = a cos θ

Back

detailed matrix form

Front

labeling the rows and columns of a matrix

Back

m

Front

stands for the amount of rows in a matrix

Back

Section 9

(50 cards)

x²=4py (parabola)

Front

+ opens up - opens down

Back

ellipse

Front

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

Back

co-vertices for horizontal ellipse

Front

(h,k±b)

Back

Formula for up/down parabola

Front

y=a(x-h)²+k

Back

Combinations

Front

-Order does not matter -nCr ~n = number of total items ~r = number of items being combined

Back

Independent Probability

Front

Probability of A and B P(A) x P(B)

Back

What happens as e approaches infinity?

Front

The conic becomes less circular and more stretched out

Back

formula for minor axis for both

Front

2b

Back

What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC < 0?

Front

Hyperbola

Back

Formula for directrix of left/right parabola

Front

x=h−1/4a

Back

focus for horizontal ellipse

Front

(h±c,k)

Back

Axis of symmetry formula for up/down parabola

Front

x=h

Back

Centers for both

Front

(h,k)

Back

formula for major axis for both

Front

2a

Back

Formula for focus of Up/Down Parabola

Front

(h,k+1/4a)

Back

x²+y²=r² (circle)

Front

the set of all points that are equidistant from a fixed point called the center. center (0,0) radius r

Back

What is the conic with an e < 1?

Front

Ellipse

Back

vertices for left/right hyperbola

Front

(h±a,k)

Back

vertices for vertical ellipse

Front

(h,k±a)

Back

Formula for left/right parabola

Front

x=a(y-k)²+h

Back

Formula for directrix of up/down parabola

Front

y=k−1/4a

Back

formula for hyperbola opens up/down

Front

Back

What happens as e approaches 0?

Front

The conic becomes more and more like a circle

Back

What is the conic with an e > 1?

Front

Hyperbola

Back

Center of circle

Front

(h,k)

Back

How to determine direction of left/right parabola

Front

If a>0, opens right. If a<0, opens left

Back

Radius

Front

r

Back

y²=4px (parabola)

Front

+ opens right - opens left

Back

parabola

Front

the set of all points that are equidistant from the focus and directrix. p= distance from vertex to focus and from vertex to directrix.

Back

vertices for up/down hyperbola

Front

(h,k±a)

Back

Formula for focus of left/right parabola

Front

(h+1/4a,k)

Back

Formula for circle

Front

(x-h)²+(y-k)² = r²

Back

Polar equation of a conic

Front

r = ep / 1 - e cos(θ)

Back

co-vertices for vertical ellipse

Front

(h±b,k)

Back

Definition of eccentricity

Front

c/a

Back

What is the conic with an e = 1?

Front

Parabola

Back

slope for asymptote for up/down

Front

±a/b

Back

formula for horizontal ellipse

Front

Back

Dependent Probability

Front

Probability of A or B P(A) + P(B) - Overlap (Possibilities that fall into A and B)

Back

vertices for horizontal ellipse

Front

(h±a,k)

Back

What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC = 0?

Front

Parabola

Back

formula for hyperbola opens left/right

Front

Back

What type of conic is of the form Ax² + Cy² + Dx + Ey + F = 0 where AC > 0?

Front

Ellipse (or circle)

Back

How to determine direction of up/down parabola

Front

If a>0, opens up. If a<0 opens down.

Back

slope of asymptote for left/right hyperbola

Front

±b/a

Back

formula for vertical ellipse

Front

Back

focus for vertical ellipse

Front

(h,k±c)

Back

hyperbola

Front

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

Back

Axis of symmetry formula for left/right parabola

Front

y=k

Back

Permutation Formula

Front

You have n items and want to find the number of ways r items can be ordered

Back

Section 10

(10 cards)

focus for up/down hyperbola

Front

(h,k±c)

Back

Asymptote

Front

A line that a graph approaches but never touches.

Back

Focus (Foci - plural)

Front

One of two fixed points at the centers of an ellipse.

Back

Major Axis

Front

An ellipse's maximum length.

Back

focus for left/right hyperbola

Front

(h±c,k)

Back

Minor Axis

Front

An ellipse's minimum length.

Back

converse axis for both

Front

2b

Back

formula for c when dealing with hyperbola

Front

Back

transverse axis for both

Front

2a

Back

Hyperbola

Front

The graph of a rational function.

Back