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# The Big Ideas of Advanced Functions

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## Esther Chi

on 18 January 2014

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#### Transcript of The Big Ideas of Advanced Functions

The Three Big Ideas Are:
1. Components of Domain & Range

2. AROC/IROC

3. Using Formulas to Solve Equations

Formulas are a great tool to help solve a variety of math problems. We looked at many over the course of the semester and learned different ones for each chapter.
3. Using Formulas to Solve Equations
A rate of change is a measure of the change in one quantity with respect to a unit change in another quantity (independent variables). There are two types, average rate of change and instantaneous rate of change.

2.AROC/IROC
Domain and range are an essential component when graphing because the domain shows the x-values of a certain graph while the range shows the y-values of a certain graph. They can also show where the function can't go on the graph too.
1. Components of Domain & Range
The Big Ideas of Advanced Functions
The three ideas cover these units in the course:

Chapter 1- Polynomial Functions

Chapter 2- Polynomial Equations and Inequalities

Chapter 3- Rational Functions

Chapter 4- Trigonometry

Chapter 5- Trigonometric Functions

Chapter 6- Exponential and Logarithmic Functions

Chapter 7- Exponential and Logarithmic Equations

Chapter 8- Combining Functions
AROC and IROC can be used for any type of function because you are finding the slope of either the secant or the tangent line.
Inequalities
Asymptotes
Trigonometric
Equations
AROC
IROC
Special Cases
Vertical Asymptotes
Horizontal Asymptotes
Discontinuities
Oblique Asymptotes
Polynomial Equations
Remainder Theorem
Factor Theorem
Cofunction Identities
Compound-Angle Formulas
Other Trigonometric Identities
Exponential and Logarithmic Functions
Power Law
Change of Base Formula
Logarithmic Scales
Product and Quotient Laws
Combining Functions
Sum and Difference
Products and Quotients
Composite
Domain and range can be found whenever there is a function and a graph which means most of the course material we learned can be used to find the domain and range. They are also a part of the key features.
Asymptotes can be useful by themselves but are also included when determining the domain and range of certain types of function. They show which x and y values the function can't be and are only found in rational, trigonometric, exponential and logarithmic functions for this course.
HA is a horizontal line on the graph where it shows which y-value the function can't be. You can determine it by taking the highest degree term in the numerator and the denominator then dividing.
This can be used after you do long division as a way to determine what the remainder is. Two ways to write this are

or
.
This is used to determine whether is a factor of a polynomial when
or if is a factor when .
These identities show the relationship between sine, cosine,tangent, etc.
Exponential functions are written in the form

while the reciprocal, logarithmic functions, are written
This can be used to solve equations with unknown exponents and is shown as
This formula is used to evaluate or graph logarithmic functions with any base.
These provide a convenient method of comparing values that have a very large or small range. There are many examples of logarithmic scales that are used in the real world.
Here are some examples of logarithmic scales:
These are used to simplify expressions and solve equations.
Product Law:

Quotient Law:

There are various ways of combining one or more functions which include sum and difference, products and quotients and composite.
The specific chapters we will be looking at are:
-Polynomial Equations
-Trigonometric Equations
-Exponential and Logarithmic Functions
-Combining Functions
The VA is a vertical line on the graph that shows what x-values the function can't be. It is determined by calculating the x-intercepts in the denominator. This is normally found in rational and trig. functions.
If the degree numerator is smaller than the denominator, HA is y=0. When they are equal, the HA is the value after dividing. Once it is bigger than the denominator, it is no longer a HA.
This occurs when a rational function is simplified which would then cause the restriction to be the x-value of the "hole". You would have to sub in this number into the function to determine the y-value of the "hole".
An oblique asymptote is a diagonal line on a graph where the function can't touch. Determining the oblique asymptote is the same as HA but results in the degree numerator being bigger than the degree denominator. The actual value of the oblique asymptote is the quotient after solving the rational functional.
The next two concepts only showed up occasionally in the course but are important when graphing certain functions.
Inequalities can show the specific group of values of x and y where the function can be and are similar to regular equations but have an inequality sign instead of an equal sign.
In the form
AROC is the slope between two points on a graph which is also called the secant line.
IROC is the slope of a certain point on a graph which is called the tangent line.
Double-Angle Formulas:
Pythagorean Identities:
Reciprocal Identities:
There were many formulas we learned for this unit and all of them contained either sine, cosine, tangent, etc.
Each of these formulas depend on two or more angles and can be developed using algebra and the unit circle.
You can combine functions together by adding or subtracting them.
You can combine functions together by multiplying or dividing them.
This method of combining functions is by substituting the second function into the x-value of the first function.
The second equation is just a different way of showing the first equation. It is called the composition of f with g.
Inequality Signs
> - this sign means greater than
- this sign means greater than or equal to
< - this sign means less than
- this sign means less than or equal to
D: {xER}
R: {yER|y < 3}

*The vertical asymptote is y=3 so none of the y-values of the function can be higher than that.
You can determine the intervals for the inequality algebraically or by using the number line. X-intercepts are used when solving inequalities.
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