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Chapter 1


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Section 1-1: The Language of Algebra



Section 1-1
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In chapter one section one, we reviewed many things. A variable is a placeholder, and algebraic expression is a combination of a variables and numbers using operators and an algebraic sentence is when a verb is used with the numbers, variables and operators. In order to evaluate an expression, you must substitute for variables and find the result. Our version of the order of operation (PEMDAS) was exchanged for GEMDAS; grouping symbols, exponents, multiplication/division left to right, and add/subtract left to right. This lead into equations or a sentence where two expressions are equal. Then formulas or a rule that states a variable equals a certain expression such as Area of a Triangle=1/2bh. These are the basics for chapter one and further in Advanced Algebra.

Shannon S.


In this lesson many things are reviewed. The first reviewed item is a variable which is a placeholder.an algebraic expression is a combination of numbers and variables using operators such as + - x / to combine them. An algebraic expression is related with "verbs" such as = > < etc. substitution is needed in order to evaluate expressions by putting numbers in the place of a letter. Rather than the outdated PEMDAS, we now have a new version called GEMDAS.The new letter G stands for grouping symbols.Exponents. Multiplication/Division.Addition/Subtraction.The new and improved operator system led to the last two vocabulary words of the lesson.Equation- a sentence where the two expressions are equal.Formula- a rule that says a variable will always equal a certain expression.
Caiti T.

Section 1-2: What is a Function?

Question Link

AA Section 1-2
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Lesson 1-2 is all about math functions. What is a function you ask? A function is simply a corresponding or pairing between 2 variables, so that each value of the independent variable corresponds to exactly one value of the dependent variable. So what exactly are dependent and independent variables? The dependent variable is the value that relies on the independent variable. Now the independent variable determines the dependent variable and which in turn the independent variable relies on nothing. Now if indeed the variables are a function then we say x is a function of y. Ok so a domain of a function is anything that can be substituted for the independent variable. That entails the range of a function which is any value that can be substituted for the dependent variable. Although the substitutions for the domain are called the input; and for the range it’s called the input and for the range it’s called the output. Ok now that we have that straightened out here are some examples of common numbers used as the domain. The first set is natural numbers such as 1, 2, 3, 4, 5, etc. Next you have whole numbers which are 0, 1, 2, 3, 4, etc. Then we have integers and they’re -2,-1, 0, 1, 2, etc. Next we have rational numbers which are represented as ratios in the form a/b, however b cannot equal 0. Last we have real numbers which are represented by decimals such as 0, 1,-7, 2.34, 3.14, etc. Now that pretty much covers all of section 1-2.

Richard S.

The focus of lesson 1-2 is on functions and variables. We learned what the difference is between independent and dependent variables in this lesson. There is a question on our worksheet asking, "Why do we call them dependent and independent variables?". The answer is simple, the independent variables were given that name, because they don't rely on anything. The dependent variables, however, rely on the independent variables to determine their value. Next we learned about functions, which are a pairing between two variables, so that the independent variable matches with only one dependent variable. We also were taught that "Is a function of" is a phrase used when working with functions. The meaning of the words domain and range were a part of this lesson as well. Domain is anything that can possibly be substituted in for the independent variable. Range is any value that can be substituted for the dependent variable. The words "input" and "output" are also used, and each another word for the independent and dependent variables. In addition to all of this, we learned about the common sets of numbers, such as the natural numbers, the integers, the whole numbers, the rational numbers, and the real numbers. Each set has their own numbers, such as the integers, which consist of numbers such as -2,-1,0,1,and 2. That takes us to the end of lesson 1-2.

Max S.

Section 1-3: Function Notation



Section 1-3
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Today in Class or August 28th, 3rd period learned about notations. There are two kinds of Notations...

A. Euler's Notation (Oiler) B. Mapping Notation (Arrow Notation)
F(x) Read as: "F of x" A:x --> Read as: "A maps x onto"
Independant Variable is inside Indepndant after colon.
parantheses.

Example: x= 4

F(x)= 2x + 38 A:x --> 5x + 7
F(4)= 2(4) + 38 A:4 --> 5(4) +7
F(4)= 8 + 36 A:4 --> 20 +7
F(4)= 44 A:4 --> 27

Dakota E.



In 1-3: Function Notation we learned about different notations.
Eulers Notation: f(x), this is read as "f of x." f(x) is the dependent variable and the variable inside the parenthesis is the independent variable.
Mapping Notation a.k.a. arrow notation: A:x->, this is read as "A maps x onto." The dependent variable would be A:x, and the independent variable would be after the colon.
In this lesson we learned that another name for the independent variable is the argument and another name for the dependent variable is the value.
Examples:
B(x)=x+5
B(2)
B(2)=2+5
B(2)=7
This would be read as "B of 2 equals 7"

F(x)=2x+1
F:2
F:2->2(2)+1
F:2->5
This would be read as "F maps 2 onto 5"

Austin L.

Section 1-4: Graphs of Functions


AA Section 1-4
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Horizontal Axis--> contains independent variables
Vertical Axis--> contains dependent variables
Vertical-Line Test--> if a vertical ine can be drawn anywhere on a graph and only touch the graph once, it is a function.
Relation--> any set of ordered pairs
Example:
(1,1),(3,65),(7,9) They are all relations.
The range of a function is the set of values of the dependent variable that result from all possible solutions for the independent variable. Each point in the coordinate plane has (x,y) coordinates. If y=f(x) then you can say that the points form (x,f(x)).


*Marissa B and Crystalene E*
= ]

Section 1-5: Solving Equations


Section 1-5
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Distributive Property - For all real numbers a, b, c(a + b) = ca = cb.
Opposite of a sum theorem**- For all real numbers a and b, -(a + b) = -a +-b.

To clear fractions you must multiply by a common denominator. If there are more than one term on each side of the equation it may be much easier to use the distributive property. When solving and equation like 55 + 2(33d-2) = 11d you would use the distributive property and distribute the 2 to 33d and -2. this would then give you 55+66d-4=11d. For the opposite of sum theorem you distribute the negative throughout the numbers inside the ().

Example: 6m - (5 -9m) = 13
6m - 5 + 9m = 13
15m - 5 = 13
15m = 18
m =18/15 = 6/5

Dominic F.


Section 1-6: Rewriting Formulas


AA Sectionn 1-6
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Lesson 1-6 is about rewriting formulas. You can solve the formula for different variables.
example;
d=rt
d/r=rt/r
d/r=t

To find t you would substitute the values for d and r.
You would then divided both sides by r as shown and then simplify the equation.
This can be helpful if you want to find a different variable than the one that it is solved for.

Brittany Kortright:]

Section 1-7: Explicit Formulas for Sequences


AA Section 1-7a
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Term-Each figure, set of numbers, or value.
Explicit Formula for the Nth Term-Allows for us to find any term in a sequence.
Sequence-A function whose domain is the natural numbers.
Subscript/index-Tells us which term in the sequence that is being dealt with
Example:
Tn=7n+1 for integers n

  • 1
To find T

12 you have to substitute 12 for n.
T12=7(12)+1=84+1=85
*Gage A. and Marisa M.*

Section 1-8: Recursive Formulas for Sequences




Recursive formula is used when you have a set of statements that has the first term and tells how the the term is related to the previous terms. Its just using the term that you solved for before. So if you have an equation T2= t2 +3 then u would take the first term and place it in that slot. Then is you have T3=t3+3 then you put the answer you got for T2 in the T3 slot. On a calculator you can also use the ANS button to find wich comes next in.
KmK

A recursive formula or recursive definition
it indicates the first term or the first few terms ands tells
how the nth term is related to one or more of the previous terms.

The recursive formula first shows the first term that is put in to find the rest of
the sequence. Then it shows the next terms from the previous terms.

Examples
t1=25
tn= 5*the previous answer+6,for int. greater than 2

t1=25
t2=5*25+6=131
t3=5*131+6=661
t4=5*661+6=672
t5=5*672+6=3366

Autumn T.

Section 1-9: Notation for Recursive Formulas




In a recursive formula it tells you the first term, how to find the nth term based on the previous term. We learned how to write the recursive formula more mathematical, by writing t n-1. It is exactly the same as writing previous term in the equation.

Example:
t1=300
tn=previous term -30, for integers greater than or equal to 2

Instead you would write
t1=300
tn=(tn-1)-30, for integers greater than or equal to 2

Then you would plug in the previous term and solve


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Elissa M. and Kaitlynn R.

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