## Math 4 Unit 4: Rational Functions

#### Graphing Rational Functions

Identify the holes, intercepts, HA, VA, and SA, then graph.

#### Multiplying Expressions

Multiply and simplify.

#### Dividing Expressions

Divide and simplify.

#### Subtracting Expressions

Subtract and simplify.

#### Solving Inequalities

Solve the inequality.

## Math 4 Unit 3: Polynomials

Completely factor x^2+6x+8

= (x+4)(x+2)

Completely factor 3x^2+4x-7

= (3x+7)(x-1)

Completely factor

= x^2(x-7) – 3(x-7)

= (x^2-3)(x-7)

#### Factoring Quartics

Completely factor 2x^4-28x^2+90

= 2(x^4-14x^2+45)

= 2(x^2-9)(x^2-5)

= 2(x+3)(x-3)(x^2-5)

#### Solving Quartics

Completely factor.

= 2x^4-28x^2+90

= 2(x^4-14x^2+45)

= 2(x^2-9)(x^2-5)

= 2(x+3)(x-3)(x^2-5)

x=-3, 3, +-sqrt 5

#### Vertex Form

Rewrite in vertex form.

#### Graphing Polynomials

• Determine the End Behavior based on the Leading Coefficient Test and write the end behavior of this polynomial in Limit Notation.
• Find the zeros of the polynomial function
• Find the y-intercept of the polynomial function
• Graph the polynomial function.

#### Write a Polynomial with Roots

Find a polynomial function with the given zeros: 0, 2 mult 2

#### Long Division

Divide with Long Division

#### Synthetic Division

Divide with Synthetic Division

#### Rational Roots Theorem

Use the Rational Roots Theorem to find all possible roots and then solve.

## Math 4 Unit 2: Functions and Their Graphs

#### Vocab

Domain – the span of all the x-values of a function

Range – the span of all the y-values of a function

x-intercept(s) – the point(s) where a function touches the x-axis

y-intercept – the point where a function touches the y-axis

Horizontal asymptote(s) – the imaginary line(s) where the domain of a rational function approaches on both sides but never touches.

• There are three rules that define where the asymptote will be and how it will look like:
• If the degree of the numerator is less than the degree of the denominator, the asymptote will, by default, be y = 0.
• The asymptote will be higher or lower if there is a k value in the function.
• If the degree of the numerator is equal to the degree of the denominator, the asymptote will, by default, be at y = (degree of the numerator/degree of the denominator)
• The asymptote will be higher or lower if there is a k value in the function.
• If the degree of the numerator is greater than the degree of the denominator, there will not be a horizontal asymptote.

Vertical asymptote – the imaginary line(s) where the range of a rational function approaches above and below but never touches.

Local minimum – the point where a positive even polynomial’s vertex is, where the function does not go any lower.

Local maximum – the point where a negative even polynomial’s vertex is, where the function does not go any higher.

End behavior – the direction, attributes, and/or limit of a function as it goes left or right.

#### Application

The height of a passenger on an escalator is modeled as

where t is the time in seconds after the passenger has boarded the escalator on the second floor and h is the height in feet above the first floor.

This means it takes the passenger 16 seconds to reach the first floor (0 feet).

This means the passenger is 12 feet above the first floor when he/she first gets on the escalator. Likewise, this also means that the second floor is 12 feet above the first floor.

#### Inverse Functions

Find the inverse of the following function:

#### Composition of Functions

Find f(g(x)) and its domain if:

and:

#### Decomposition of Functions

Find what h(x) decomposes into if:

and:

## Math 4 Unit 1: Sequences and Series

#### Recursive Sequence

A sequence expressed in the form of an operation that is performed on the previous term of the sequence to find the next. To use a recursive sequence, the previous term must be given to calculate any following terms.

Example: Find the first 4 terms of the recursive sequence below.

#### Explicit Sequence

A sequence expressed in the form of an equation in terms of n, where n is the term’s position in the sequence. This method of expressing a sequence does not require any previous terms to find following terms; it only requires you have the position of the term (n) you are looking for and the first term of the sequence.

Example: Find the first 4 terms of the explicit sequence below.

#### Arithmetic Sequence

Example: Find the 4th term of an Arithmetic sequence with a first term of 7 and a common difference of -4.

#### Arithmetic Series

Example: Find the sum of the first 10 terms of an Arithmetic sequence with a first term of 7 and a common difference of -4.

#### Geometric Sequence

Example: Find the 6th term of a Geometric sequence with a first term of 2 (a1) and a common ratio of 3 (r).

#### Geometric Series

Example: Find the 6th term of a Geometric series with a first term of 2 (a1) and a common ratio of 3 (r).

#### Application Problem

A cybersecurity technician has a starting salary \$106,000. Per his contract with the company, he is given a 5% raise each year. If he works for 20 years, how much will he have earned total by the end?

a1 = 106,000 because the salary starts at \$106,000

r = 105% or 1.05 because the salary increases by 5% each year

n = 20 because the technician is working 20 years

#### Infinite Sequences

Find the sum of the following sequence, if there is one:

## 1/20, 1/40, 1/80…

#### Convergent & Divergent Series

A convergent series has an r value with an absolute value less than 1, meaning the sequence will get closer and closer to 0 as it progresses. This means you can calculate the sum of the series to the infinite term, as there will be a finite sum.

For example, the geometric series in the previous question has an r value of 1/2 (|1/2|<1), because its terms are getting closer and closer to 0 (in this case, smaller and smaller)

On the other hand, a divergent series has an r value with an absolute value greater than 1, meaning the sequence will get farther and farther away from 0 as it progresses. Because of this, you cannot find the sum to an infinite term as the sequence has no limit to how large its terms it can get.

For example, this geometric series has an r value of 2 (|2|>1), which means the terms will get larger and larger, farther and farther away from 0, meaning there is no real way to find an infinite total of the series since there is no limit to how high it will go.

#### Summation Notation

Find the sum using the correct sequence formula.

#### Binomial Theorem (all terms)

Expand completely.

#### Binomial Theorem (single term)

Find the 3rd term in expansion of