~Precalculus!~

Precalculus is not as scary as it may seem. Do not worry. We got you :D

For the trigonometry formulas, please refer to the Trigonometry Page

Many concepts are repeated from Algebra 2, so they may seem familiar to you

Domains and Graphing Basics

Coordinate Plane Quadrants

Distance Formula

Finds the distance between two points on a coordinate plane
The distance formula is also the pythagorean therom.
The distance between two points is the hypotenuse of a right triangle.

Midpoint Formula

Finds the middle point in between two points on a coordinate plane
Dividing by two finds the center x and y point

Intercepts

x-intercepts: let y = 0 and solve for x
y-intercepts: let x = 0 and solve for y
PLEASE write intercepts in coordinate form! Ex. (0, 5)

Symmetry

x-axis Symmetry

One half of the graph is the same as the other half
Opens horizontally
Even exponent on the y value
Ex. x = y⁴ - 3

y-axis Symmetry

One half of the graph is the same as the other half
Opens vertically
Even exponent on the x value
Ex. y = x² + 4

Origin Symmetry

Replacing y with -y and x with -x creates the same equation
Most likely has an odd exponent
Ex. y = x⁵

Equation of a Circle

Center = (h,k)
Radius = r
Find the midpoint of the circle using two points to find the center
Distance from a point and the center is the radius

Slope

AKA average rate of change

Slope-Intercept Form

m = slope
b = y-intercept

Point Slope Form

The same as Slope-Intercept Form
Use a point and the slope to find the slope-intercept form of the slope

Standard Form

Can convert into slope-intercept form

Parallel & Perpendicular Lines

Parallel lines

Both slopes have the same slope (m). Only difference is the y-intercept
Ex. y = 2/3x + 1 and y = 2/3x + 4

Perpendicular lines

Both slopes have an opposite value and reciprocal of each other
Ex. y = 2/3x + 1 and y = -3/2x - 7

Interval Notation

(_, __)

The domain DOES NOT include the x values
If the coordinate is part of a graph, leave a hole in the coordinate

[_, __]

The domain includes the x values

(_, _] or [_, __}

The respective x values will be included/not included

Crit Numbers

Values of x that make the equation = 0 or undefined
Used when figuring out the domain of a function
Create a number line that shows the restricted domains for the function.

Even and Odd Functions

Even: f(-x) = f(x) (not a function)
Odd: f(-x) = -f(x) (origin symmetry)
Odd: -f(-x) = f(x) (origin symmetry)

For all Common to the Domains of f and g

Inverse Functions

IS NOT 1/f
Swap x and y then solve for y to find the inverse function
If you are trying to find the inverse of a parabola, make sure to find the restriction

Keep in mind... Restrictions

Restriction: prevents the graph from going beyond a part of the graph
Reason: If an x value goes beyond the restriction, the y value will be no solution in the original function
Ex. x >= -2 prevents the graph from going lower than (-2, y)

Ex. The inverse of f(x) = (√x + 1) - 2 is f(x) = (x+2)² - 1.
The restriction is x > -2. If x = -2, the original equation will be √-1 - 2 which is no solution

Composition of functions

A function becomes the x of another function
Can be written in this notation

Polynomials

First equation is in quadratic form, second one is in vertex form

Parabola Functions

Vertex in quadratic form: (-b/(2a), f(-b/(2a)))
Vertex in standard form/vertex form: (h, k) (when k = y and h = x)

Polynomials of the Higher Degree (Exponent)

When the leading coefficient's exponent is even, it is a symmetrical function
When the leading coefficient's exponent is odd, it is an odd function
End behavior: graph goes up/down
Exponent - 1 = turning points (amount of times the graph changes from up to down or vice versa)

Multiplicity: how many times each value of x is multiplied in an equation

Intermediate Theorem

Goal: find the rational 0s of a polynomial
Possible rational 0: values of x that could make the equation equal to 0
If a possible rational 0 provides a negative remainder and another provides a positive remainder, the actual rational 0 will be a value in between them

Division of Polynomials

Long Division

For dividing equations such as x³ - 1, remember to include the missing degrees (2 and 1)
DO NOT forget to use the remainder in the answer. Put the remainder over the divisor in fraction form and add it to the final answer

Synthetic Division

Combining Synthetic Division and Intermediate Theorem

Imaginary numbers

Complex Conjugates

If a + bi is a zero, then and a - bi is a zero
Ex. the complex conjugate of 2 - 3i is 2 + 3i. They are both zeros
Multiply by the complex conjugate of an imaginary fraction to simplify it

Descartes Rule of Signs

Trick to find the amount of real 0s in a polynomial function
Number of sign changes in the function - 2 = number of POSITIVE real 0s
Number of real 0s could be the amount of sign changes as well
f(-x) sign changes in the function = number of NEGATIVE real 0s

Asymptotes of Rational Functions

an image of a rational function with a hole in it

Holes

Place where x has no y value because it is undefined
If the same binomial is in the numerator and denominator, the x value is the x for the hole. The binomials cancel out.
Plug in the x value into the remaining equation to find the y value of the hole

Vertical asymptote

X value that makes the denominator 0 (binomial that is not the hole)

Horizontal asymptote

HA: horizontal asymptote
If the numerator's leading degree > denominator's leading degree, HA: y = 0
If the numerator's leading degree = denominator's leading degree, HA: y = num degree / den degree
If the numerator's leading degree < denominator's leading degree, there is no HA

Slant Asymptote

Quotient of polynomials without the remainder
Use synthetic division

Logarithms, Exponentials, and Compound Interest

Exponentials

Has a horizontal asymptote
Focal point: first integer point closest to HA
a: distance between focal point and horizontal asymptote (+ = focal point is above HA, - = focal point is below HA)

Logarithms

Has a vertical asymptote
logX: log₁₀X
lnX: log_eX
log_an = x is the same as a^x = n
log_an = x is the same as ln n / ln a = x

Compound Interest

For n compoundings per year

P = principal/initial value for investment
r = annual interest rate
t = years
n = number of compoundings per year

Continuous Compounding

Exponential growth/decay

Speed and Revolutions

Radians and Degrees

Degrees

θ = theta
Positive angles: counterclockwise
Negative angle: clockwise (add 360 to find the positive angle)
Coterminal: two different angles are in the same position on the coordinate plane

Radians

r = radius
s = distance between angle 0 and angle θ
One rotation (circumference of a circle): C = 2πr

Speed

Linear Speed

Total DISTANCE the object moving in a circular arc will go
t = time
s = length of the arc

Angular Speed

w = omega
Total SPEED the object moving in a circular arc will go

Number of Revolutions

2π = one revolution around a circle

Linear Speed of a Moving Object

Vectors

Magnitude

Click here for the distance formula

The same formula as the Distance Formula
The length/distance between the initial point and terminal point
||v||

Standard Position

Initial point is at (0,0)
Component form: terminal - initial <v₁, v₂> (initial point is now (0,0))
Component form for a vector in standard position: <terminal point>

Finding the magnitude in component form (same as the distance formula):

Unit Vectors

Finds a vector with a magnitude of 1 that goes in the same direction as the vector used in the formula
V = component form of a vector
||v|| = the magnitude of a vector
Multiply the unit vector by a number to get a vector with a specific magnitude in the same direction as V

Linear Combination of i and j

<v₁, v₂> → v₁i + v₂j
Direction angle:

If asked to find the component form of a vector given the magnitude and direction,
make a right triangle with the vector and use sin and cos to find the x and y of the terminal point
i = cos (horizontal) j = sin (verticle)

Resultant

Resultant of u and v is u + v

Dot Product

u · v = u₁v₁ + u₂v₂
Angle between vectors u and v:

Orthogonal Vectors: u · v = 0

Solving Systems of Equations with Matrices

Matrix

Augmented Matrix

Matrices

Way to express a system of linear equations. Only the coefficient in present
Augmented Matrices: A matrix with the answer to each system on the right side. It is separated by 3 vertical dots or a line.
Whenever you change one side, you change the other.

Forms for Matrices and Systems of Linear Equations

Row-Echelon Form: x, y, and z, have a coefficient of 1. Solve by back substitution.
Gaussian Elimination = row-echelon form

Reduced Row-Echelon Form: x, y, and z are already clearly solved and have a coefficient of 1.
Gauss-Jordan Elimination = Reduced Row Echelon Form

Back Substitution: Plug in z in the y equation then plug in z and y into the x equation

Inverse of a Square Matrix

Make an augmented matrix where the right side is equal to a reduced row echelon form
Make the left hand side into a reduced row echelon form.
The right hand side will be the inverse matrix
Whatever you change on one side, you change the other

Determinants

2x2

Cross multiply a · d and b · c
Subtract the product of ad from bc

Terms (needed for 3x3 and greater)

Refer to this section for the 3x3 section
Minor: determinant of a matrix with a row and column crossed out
Cofactor: a minor that changes signs depending on where it is in the original matrix. Refer to the image above.

3x3

Pick one row of the matrix to cross out
Make 3 2x2 matrices by crossing out one different column for each matrix. The 4 remaining values are the 2x2 matrix
Solve for each matrix's determinant (minors). You will have 3 minors
Change the 3 minors to cofactors accordingly
Add all the cofactors together to get the determinant of the matrix

Cramer's Rule

Way to find the x and y of a system of linear equations
Find the determinant of D, Dx, and Dy then use the formulas above to find x and y (x, y)

Probability

Factorials

Represented by n!
Multiply by every number leading up to n
Ex. 5! = 1 · 2 · 3 · 4 · 5
Used for combinations and permutations

Notes

Use these notes when assessing how to solve the probability of a scenario
Adding (+): you have a choice/or
Multiplying (·): calculate probability/and
n = total amount of items in a scenario (refer to the two images below)
r = amount of positions/number of objects to choose (refer to the two images below)

Permutation

Use permutations when the order in a scenario matters
Ex. How many ways can a group of people come in 1st, 2nd, and 3rd place

Combination

Use combinations when the order in a scenario DOES NOT matter
Ex. Pick 4 meals out of a menu with a total of 15 meals

Sample Space

Set of all possible outcomes

Probability of an Event (fraction/percentage)

n(e) = the event
n(s) = sample space
p(e) = probability of an event
Ex. What is the probability of flipping heads on a coin? 1 (event)/2 (total possible outcomes: heads or tails)

Mutually Exclusive Events

Use addition to solve the probability of the event.

Two events from the same sample space have no common outcomes

Independent Events

Use multiplication to solve the probability of the event.

2 events are independent. The occurance of one event has no effect on the other

Arithmetic and Geometric Sequences

n

Represents the term of a number in a sequence. Essentially where the number is in a sequence.

Summation Notation

Easy way to write a summation of a sequence
n = the amount of times the right side is added to itself (represents the last term of the sequence)
i = contributes to the amount of times the right side is added to itself (represents the first term of the sequence)
If i does not equal 1, the sequence starts at another place in the sequence

Infinite Sequence

A sequence that goes on infinitely. n = infinity
Partial sum: a limit (n) is set on the infinite sequence.
Infinite series: sum of all terms in an infinite series

Arithmetic sequences

Finds the nth term of a sequence. Has LINEAR nth term rules (addition/subtraction).
d = common difference (pattern in the sequence). Ex. add 6 for each term
The bottom formula is the same as point slope form

Used to find d in an arithmetic sequence. You can use d to find the arithmetic equation of the sequence
Same formula used to find the slope of a line
Numerator: subtract the two numbers
Denominator: the subtraction of the nth term of both numbers

Efficient way to find the sum of the first n terms of an arithmetic sequence

Geometric Sequences

Finds the nth term of a sequence. Has EXPONENTIAL nth term rules (addition/subtraction).
r = common difference (pattern in the sequence). Ex. add 6 for each term

Finds the sum of the first n terms of a geometric sequence

Finds the sum of an infinite geometric series

Binomial Theorem

Use the triangle above to expand a binomial by its respective exponent
Multiply each term in the expansion according to the triangle
First row: exponent = 0
Second row: exponent = 1
Third row: exponent = 2
Fourth row: exponent = 4. etc.

Example

Expand (2x + 3y) ⁴
Binomial is to the power of 4
1(2x)⁴ (3y) ⁰ + 4(2x)³(3y)¹ + 6(2x)²(3y)² + 4(2x)¹(3y)³ + 1(2x)⁰(3y)⁴
Simplify

Calculus

Mathematical Induction

Used for calculus. Plug in the formulas below accordingly to solve the summation.

i

i ²

i ³

Riemman Sum

Used to find the area between a function and the x axis given domain [a, b]
[a, b] = the a and b that is used to find ∆x

Limits

L = limit
c = x value of the limit (x approaches c, the x value of the limit)
f(x) = equation of the graph
0/0 = indeterminant form. If f(x) divides by 0 by plugging in c into the equation, rearrange the equation to not get an answer in indeterminant form

When the Limit Doesn't Exist

Top left: step discontinuity
Top right: infinite discontinuity
bottom: oscilating behavior

Derivatives

Use this formula when you want to find the derivative of a function
At the end, plug substitute h with whatever value it is approaching. The answer is your limit
An easier way to find derivatives is in calc AB and calc BC, however, this is how it is taught in precalculus

Equation of the Tangent Line

Definition of a derivative at a point x = c

Same as the derivative formula
x will usually be provided when solving for the derivative
Point of tangency: x = the x/c/point provided. y = f(x)
Use point slope form to find the tangent line. The slope (m) = the limit/derivative. The x and y values are the point of tangency

Conic Sections

Parabolas

p = distance between the vertex and focus point
h = changes the x position of the vertex
k = changes the y position of the vertex
The directrix is always behind the parabola's vertex
The focus is always directly in front of the vertex

Ellipses

h = changes the x position of the center
k = changes the y position of the center
a = distance between the center and vertex
b = distance between the center and co-vertex
c = distance between the center and focus (the focus is directly in front of the vertex)

Hyperbolas

h = changes the x position of the center
k = changes the y position of the center
a = distance between the center and vertex
b = distance between the center and co-vertex
c = distance between the center and focus (the focus is directly in front of the vertex)
There are two asymptotes that cross each other at the center

Parametrics

Represent the direction an object travels
The direction of a parametric always goes in the direction of where t increases
t = parameter