Transcript Language of Algebra

LANGUAGE OF ALGEBRA
•
To be able to write an English paper you’re going to have to
know just some basic rules of language. For instance, you have
to obviously know things such as the alphabet and punctuation.
•
Likewise, mathematics also has a set of rules, definitions, and
structures that must be followed.
•
Many students struggle with math because they believe it is
very disjoint and isolated; that it is just a bunch of formulas
that have to be memorized.
LANGUAGE OF ALGEBRA
• The objective of this refresher is to hopefully encourage students
to see the patterns and structures of math.
• Math is highly logical and process oriented. If students can
recognize this and begin classifying topics and problems rather
than just trying to memorize a soup of information then they
inevitably will succeed in their math courses.
• In this refresher we will cover 9 math concepts and skills that
are vital to any math class and will ideally encourage students to
begin compartmentalizing problems on their own instead of
attempting to memorize everything.
9 CONCEPTS
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Number Types
Functions
Transformations
Factoring
Distributing
Fractions
Exponents
Solving Quadratic Equations
Logarithms and Exponentials
NUMBER TYPES AND CLASSIFICATIONS
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Natural Numbers – The counting numbers 1, 2, 3, 4, 5, … are called the natural
numbers. Sometimes, but not always, they include 0. They are typically used for
counting whole objects or elements. (e.g. the population of the Earth, 5 fingers on
a hand) They are denoted as ℕ or a boldface N.
•
Integers – The integers {… , −3, −2, −1, 0, 1, 2, 3 … } are just the natural numbers,
their respective negative values, and 0. They are typically used for values that can
be positive or negative. (e.g. loans and debts, temperatures). They are denoted as
ℤ or a boldface Z.
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Rational Numbers – Pick any two integers from ℤ, a and b. Then any fraction for
7
3
2
𝑎
𝑏
b ≠ 0 is a rational number. For instance then, , − , , 0 etc. are all rational
11
89 1
numbers. They are denoted as ℚ or a boldface Q.
NUMBER TYPES AND CLASSIFICATIONS
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Irrational Numbers - The irrational numbers are those that cannot be expressed
as a ratio of two integers. Examples are 2 and the square roots of many other
numbers, and special numbers like e and 𝜋. Irrational numbers have no exact
decimal equivalents. To write any irrational number in decimal notation would
require an infinite number of decimal digits. (.6 is not irrational. Why?)
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Real Numbers – The real numbers are all the rational numbers and all the
irrational numbers. In short, they are all numbers from negative infinity to positive
infinity (−∞, ∞). They fill up a number line entirely. Something like time is
typically defined on the positive real numbers. (why?). They are denoted as ℝ or a
boldface R.
NOTICE: The naturals are within the integers, the integers are within the rationals, and
the rationals are in the reals. Therefore the natural and integers are also in the reals.
Mathematically: ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ.
to R.)
( N belongs to Z which belongs to Q which belongs
NUMBER TYPES AND CLASSIFICATION
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Imaginary Numbers – Imaginary numbers are the result of the square root of a
negative number. They are denoted with a lowercase “i” next to them. For
example, −1 = 𝑖 𝑎𝑛𝑑 −9 = 3𝑖.
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Often times a problem will ask you to “find all real solutions”. What does this
mean?
𝑥 − 3 2 = −49
𝑥 − 3 = −49 = ±7𝑖
𝑥 = ±7𝑖 + 3
•
Because 7i+3 is not a real number (it is what we call complex because it has a
real number (3) and an imaginary number (7i)) we would conclude that the
question has no real solutions. The imaginary numbers are typically denoted as ℂ
or a boldface C.
NUMBER TYPES AND CLASSIFICATIONS
Here are some problems for you to work out on your own:
Classify each number:
1. . 𝟖
2. 𝝅
3. −𝟏𝟎. 𝟐
4.
−𝟒𝟗
5. 𝟏, 𝟎𝟎𝟎𝒊
6. 𝟓𝟒
7. −𝟒
Answers:
1) rational
2) irrational
3) rational
4) imaginary
5) imaginary
6) natural
7) integer
FUNCTIONS
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What is a function?
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Mathematically: In mathematics, a function is a relationship between a set of
inputs and a set of outputs with the property that each input is related to exactly
one output.
•
In short: A function takes an input value and shoots out an output value. It can
only shoot out one output for each input.
• We can think of a function as a machine along an assembly line. We input a few
nails and bolts in one side of the machine and it outputs a car at the other end.
FUNCTIONS
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Functions can be defined in lots of different ways.
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Just about every function we see would look something like this:
𝑓 𝑥 = 2𝑥 + 1
 The name of our function is 𝑓 and by notation it is assumed its input values are x.
Functions like this are easy to graph on an XY plane.
 It is REALLY important that you always make the connection between a function
and its graph in your head for every single function you ever see.
FUNCTIONS
• Functions have domains and ranges.
• Domains – A function’s domain is a set of values that are allowed to be placed into
our function. They are the input values that can be plugged in. Most functions have
infinite domains. That is, any number can be plugged into the function.
• Where we see issues are for functions like :
𝑓 𝑥 =
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1
𝑥
or
Why do we have domain problems here?
g 𝑥 = 𝑥.
FUNCTIONS
Example: Find the domain of the following function
2𝑥 − 6
𝑔 𝑥 =
8𝑥 − 4
We cannot plug in any value of x into g and get an output. This is because we have a
denominator with a variable in it. We will then set the denominator equal to 0 and see
what values of x make this happen. This will be the value of x where our domain does
not exist.
8𝑥 − 4 = 0
8𝑥 = 4
1
𝑥=
2
Therefore we would write our domain as:
1
1
𝐷: −∞,
∪
,∞
2
2
FUNCTIONS
Example:
𝑓 𝑥 =
2𝑥
𝑥−8
Two things to notice here. First, we know the denominator must never be 0. However,
we also have a square which we know can never be negative. Thus, the square root
dominates, so to speak, and we set what’s inside the square root to be greater than
0, not including 0. Even though we can take the square root of 0 we cannot divide by
0.
𝑥−8>0
𝑥>8
Our domain is then: 𝐷: 8, ∞
FUNCTIONS
• Range – The range of a function is the set of values that a function can assume or
take on.
• It’s all possible 𝑓 𝑥 𝑜𝑟 𝑦 values. For instance, let us think of the graph of 𝑓 𝑥 =
𝑥. What does its graph look like? What is its range?
FUNCTIONS
Here our some problems for you to work out on your own.
Find the domain of the following functions:
1. 𝒎 𝒙 = 𝟑𝒙 − 𝟒
2. 𝒂 𝒙 = 𝒙𝟑
𝟑𝒙
3. 𝒕 𝒙 = 𝟔𝒙−𝟏𝟕
4. 𝒉 𝒙 = 𝟑𝒙 − 𝟕
𝒙
5. 𝒚 𝒙 = 𝒙−𝟗
Answers: 1) −∞, ∞
2) (−∞, ∞)
3) −∞,
𝟏𝟕
𝟔
∪
𝟏𝟕
,∞
𝟔
4)
𝟕
,∞
𝟑
5) [𝟎, 𝟗) ∪ 𝟗, ∞
TRANSFORMATIONS
TRANSFORMATIONS
Function transformations are a way to manipulate the graphs of basic functions,
producing, a similar looking , but different graph.
If our original function is 𝑓 𝑥 then the transformed function 𝐹(𝑥) will look something
like this:
𝐹 𝑥 = −𝑎𝑓 −𝑏𝑥 + 𝑐 + 𝑑
When dealing with transformations we use order of operations. We first look inside
the parenthesis. These will be horizontal changes.
We then look outside the parenthesis. These will all be vertical changes.
TRANSFORMATIONS
𝐹 𝑥 = −𝑎𝑓 −𝑏𝑥 + 𝑐 + 𝑑
Inside f(x):
Step 1 – The c term will give the horizontal shift.
Step 2 – If there is a “-” sign then we will have a horizontal reflection about the y-axis.
Step 3 – The b term gives a horizontal stretch/compression.
Outside f(x):
Step 4 – The “-” will give us a vertical reflection about the x axis.
Step 5 - The a term will produce a vertical stretch/compression.
Step 6 – The d term will give rise to a vertical shift.
TRANSFORMATIONS
Example: Sketch the graph of 𝑓 𝑥 = − 𝑥 − 2 + 4 .
Here our base function is 𝑥. This graph looks like:
TRANSFORMATIONS
We next look for what’s inside our base function: That will be 𝑓 𝑥 = 𝑥 − 2 (green).
This will result in a shift 2 units to the right.
TRANSFORMATIONS
We now look outside the base function. We first look at the negative out in front. This
will result in a vertical reflection about the x-axis. 𝑓 𝑥 = − 𝑥 − 2. (orange)
TRANSFORMATIONS
Finally we account for the vertical shift. 𝑓 𝑥 = − 𝑥 − 2 + 4(black)
TRANSFORMATIONS
The domain and range of the base function 𝑓 𝑥 = 𝑥:
𝐷: 0, ∞
𝑅: [0, ∞)
The domain and range of our new function 𝑓 𝑥 = − 𝑥 − 2 + 4
𝐷: [2, ∞)
𝑅: (−∞, 4]
How could we find the domain algebraically?
FACTORING
Factors – two numbers or terms that can be multiplied together to get another
term or number.
• 2 and 3 are factors of 6.
• 𝑥 − 3 𝑎𝑛𝑑 2𝑥 − 6 are factors of 2𝑥 2 − 12𝑥 + 18. (Is that clear as to why?)
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The process of factoring is the process of finding factors of larger terms. In
algebra, trigonometry, and pre-calculus the most common kind of factoring will
arise from problems such as
𝑓 𝑥 = 3𝑥 3 − 27𝑥 2 + 27𝑥
FACTORING
𝑓 𝑥 = 3𝑥 3 − 27𝑥 2 + 27𝑥
•
First Step: Find the largest common coefficient (number) factor in EACH term.
• 9 goes into 27 (the last two terms) but it doesn’t go into the first term, 3.
• It would appear as if 3 is the highest common factor of each three terms.
𝑓 𝑥 = 3 𝑥 3 − 9𝑥 2 + 9𝑥
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Second Step: Pull out the lowest order variable that each term has.
• In this case the lowest order is 1 so we will pull out an x.
𝑓 𝑥 = 3𝑥(𝑥 2 − 9𝑥 + 9)
FACTORING
Here are some problems for you to work out on your own.
Factor the following functions:
1. 𝟔𝒙𝟑 + 𝟐𝒙𝟐 + 𝟒𝒙
2. 𝒙𝟐 − 𝒙 − 𝟕𝟐
3. 𝒙𝟑 ∙ 𝒙 − 𝟐 ∙ 𝒛 + 𝒙 ∙ 𝒛 ∙ (𝒙 − 𝟐)
Answers: 1) 𝟐𝒙(𝟑𝒙𝟐 + 𝒙 + 𝟐)
2) (𝒙 − 𝟗)(𝒙 + 𝟖)
3) 𝒙𝒛 𝒙 − 𝟐 𝒙𝟐 + 𝒙 − 𝟐
DISTRIBUTION
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Distribution is just reverse factoring. It undoes what we factored.
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The key thing to remember when distributing is that each term within a factor has
to multiply all of the other terms in every other factor.
𝑓 𝑥 = 2𝑥 2 𝑥 − 4 𝑥 3 − 9
= 2𝑥 3 − 8𝑥 2 𝑥 3 − 9
= 2𝑥 6 − 18𝑥 3 − 8𝑥 5 + 72𝑥 2
DISTRIBUTION
Example: Distribute he following function:
𝑓 𝑥 = 3𝑥 𝑥 − 6 𝑥 − 4
I will choose to distribute the second and third terms first. There’s no good reason for
this, just a personal preference. The first two terms can be distributed first just as
well.
𝑓 𝑥 = 3𝑥 𝑥 − 6 𝑥 − 4
𝑓 𝑥 = 3𝑥 𝑥 2 − 4𝑥 − 6𝑥 + 24
𝑓 𝑥 = 3𝑥 𝑥 2 − 10𝑥 + 24
𝑓 𝑥 = 3𝑥 3 − 30𝑥 2 + 72𝑥
DISTRIBUTION
Here are some problems for you to work out on your own.
Distribute the following expressions:
1.
𝒙𝟑 − 𝒙𝟒 𝒙 − 𝟒
𝟑
𝟐
2. 𝒙 𝐱 − 𝟒 + 𝐱 𝐱 − 𝟓
3. 𝒙𝒚𝒛 𝒙𝟐 𝒚 − 𝟏
4. 𝒙𝟐 𝒙 − 𝝅
Answers: 1) −𝒙𝟓 + 𝟓𝒙𝟒 − 𝟒𝒙𝟑
𝟓
𝟑
2) 𝒙𝟐 − 𝟒𝐱 𝟐 + 𝐱 𝟐 − 𝟓𝐱
𝟑
3) 𝒙𝟑 𝒚𝟐 𝒛 − 𝒙𝒚𝒛
4) 𝒙𝟑 − 𝝅𝒙𝟐
FRACTIONS
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Often times in algebra we will have to add, subtract, multiply, and divide fractions
with variables in them. No need to panic, all of our rules for combining fractions
with only numbers stay the same.
𝑥2 + 4
𝑥−9
+ 3𝑥
𝑥−2
7
2
𝑥 + 4 3𝑥(𝑥 − 9)
=
+
𝑥−2
7
2
2
𝑥 + 4 3𝑥 − 27𝑥
=
+
𝑥−2
7
2
7 𝑥 + 4 + (𝑥 − 2)(3𝑥 2 − 27𝑥)
=
7(𝑥 − 2)
2
3
2
7𝑥 + 28 + 3𝑥 − 27𝑥 − 6𝑥 2 + 54𝑥 3𝑥 3 − 26𝑥 2 + 54𝑥 + 28
=
=
7𝑥 − 14
7𝑥 − 14
FRACTIONS
Here are some problems for you to work out on you own:
Simplify the following fractions:
1.
2.
3.
𝒙𝟐 −𝟒
𝒙𝟐 +𝟒
+
𝒙−𝟔
𝒙−𝟔
𝟒
𝒙
−
𝒙
𝟒
𝟑𝒙
𝟐
+ 𝟐
𝟐
𝟒𝒙 +𝟏
𝟒𝒙 −𝟏
Answers: 1)
𝟐𝒙𝟐
𝒙−𝟔
2)
𝟏𝟔−𝒙𝟐
𝟒𝒙
3)
𝟏𝟐𝒙𝟑 +𝟖𝒙𝟐 −𝟑𝒙+𝟐
𝟏𝟔𝒙𝟒 −𝟏
EXPONENTS
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Many times in word problems, variables with exponents pop up. That’s why its
necessary to be able to simplify these expressions.
•
The basic rules for simplifying exponents are as followed (m and n are just
numbers):
• 𝑥 𝑚 𝑥 𝑛 = 𝑥 𝑚+𝑛
•
𝑥𝑚
𝑥𝑛
= 𝑥 𝑚−𝑛
𝑛
= 𝑥𝑛𝑦𝑛
•
𝑥𝑦
•
𝑥 𝑛
𝑦
𝑥𝑚 𝑛
•
𝑥𝑛
= 𝑦𝑛
= 𝑥 𝑚𝑛
EXPONENTS
Example: Simplify the following expression:
27𝑥 7 𝑦𝑧 4
3𝑥𝑦𝑧 8
=
1
7−1 2−1 4−8
9𝑥 𝑦 𝑧
=
1
6 −2 −4
9𝑥 𝑦 𝑧
9𝑥 6
=
𝑦𝑧 4
EXPONENTS
Here are some problems for you to work out on your own.
Example: Simplify the following expressions:
𝟑𝒙𝟐 𝒚𝟒
1. 𝒙𝒚
𝟏𝟎𝒛𝒙𝒚𝒘
2. 𝟓𝒛𝒙𝒚𝒘
𝒛𝒚𝒙𝟑 𝒙
3. 𝒛𝟐𝒚𝟒𝒙
Answers: 1) 𝟑𝒙𝒚𝟑
2) 𝟐
𝟓
𝒙𝟐
3) 𝒚𝟑 𝒛
SOLVING QUADRATIC EQUATIONS
•
In many real world problems quadratic equations pop up that have to be solved.
There are two main ways in which we can solve quadratic equations.
•
The first way is through general factoring and the second is through the quadratic
formula. Something to keep in mind is that both ways will always work. Just, most
of the time, one way will be much easier than the other.
•
Let’s say we’re given the equation below and asked to solve for x. We will do so in
both of the ways mentioned above.
𝑥 2 + 𝑥 − 12 = 0
SOLVING QUADRATIC EQUATIONS
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First Way: Factoring
𝑥 2 + 𝑥 − 12 = 0
= 𝑥−3 𝑥+4 =0
This is the factored form of our original equation. Because it is set equal to 0, we have
to solve for each factor independently, or by itself.
𝑥−3=0
𝑥+4=0
→ 𝑥 = 3, −4.
Our equation then has two solutions. Namely, 3 and -4. If we were to plug them into
the equation they both would solve it.
SOLVING QUADRATIC EQUATIONS
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Second Way: Quadratic Formula
If we have the quadratic equation in the standard form:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Then the quadratic formula solves for x (every time!):
−𝑏 ± 𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
SOLVING QUADRATIC EQUATIONS
Our problem was:
𝑥 2 + 𝑥 − 12 = 0
Because we wish to use the quadratic formula we need to find a, b, and c.
𝑎 = 1, 𝑏 = 1, 𝑐 = −12
Plugging this into the quadratic equation yields:
−𝑏 ± 𝑏 2 − 4𝑎𝑐 −1 ± 12 − (4)(1)(−12)
𝑥=
=
2𝑎
2(1)
−1 ± 49 −1 ± 7
𝑥=
=
2
2
Again, we have two equations to solve:
−1 + 7
−1 − 7
𝑥=
𝑎𝑛𝑑 𝑥 =
2
2
SOLVING QUADRATIC EQUATIONS
−1 + 7
−1 − 7
𝑎𝑛𝑑 𝑥 =
2
2
6
8
𝑥 = 𝑎𝑛𝑑 𝑥 = −
2
2
𝑥=
𝑥 = 3, −4
•
Notice that these two answers are exactly the same as when we solved the
problem using factoring.
SOLVING QUADRATIC EQUATIONS
Here are some problems for you to work on our on your own.
Example: Solve the following quadratic equations:
1. 𝒙𝟐 + 𝒙 − 𝟐
2. 𝒙𝟐 − 𝟓𝐱 + 𝟔
3. 𝟐𝒙𝟐 + 𝟔𝒙 + 𝟏
4. 𝒙𝟐 + 𝟕𝒙 − 𝟖
Answers: 1) 𝒙 = 𝟏, −𝟐
2) 𝒙 = 𝟐, 𝟑
3) 𝒙 =
−𝟔±𝟐 𝟕
𝟒
4) 𝒙 = 𝟏, −𝟖
LOGS AND EXPONENTIALS
An exponential function is any function of the form:
𝑓 𝑥 = 𝑎 𝑥 , 𝑤ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟
A logarithm is the inverse of the exponential.
What this means is that if we have an exponential 𝑎 𝑥 = 𝑦 → log 𝑎 𝑦 = 𝑥.
The most common exponential is 𝑒 𝑥 . The inverse for the exponential :
𝑒 𝑥 = 𝑦 → ln 𝑦 = 𝑥
ln is referred to as the natural log and its base, or “a” term, is e. (e ≅ 2.71)
LOGS AND EXPONENTS
The graph of the exponential (red) and logarithmic (blue) functions looks like:
(remember they’re inverses!)
LOGS AND EXPONENTS
Exponential Functions:
𝐷: −∞, ∞
𝑅: 0, ∞
Logarithmic Functions:
𝐷: 0, ∞
𝑅: −∞, ∞
Let’s now use our transformation rules for the graph of 𝑓 𝑥 = −𝑒 𝑥 + 2
LOGS AND EXPONENTS
Our base function (red) is 𝑓 𝑥 = 𝑒 𝑥 . We first account for the negative out in front
which will flip the new function (blue) across the x-axis.
LOGS AND EXPONENTS
Finally, we account for the vertical shift up (orange )in 𝑓 𝑥 = −𝑒 𝑥 + 2:
LOGS AND EXPONENTS
Our new domain and range will be:
𝐷: −∞, ∞
𝑅: (−∞, 2)
How can we find our x and y intercepts algebraically now?
RECAP
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If you are still a little fuzzy about a particular topic or would just like some more
problems to work out for review, on the CAPS webpage there are many more
detailed reviews for all of these topics and others as well with practice problems.
Here are some online resources that are very helpful as well:
Khan Academy
Virtual Math Lab
Purple Math
Paul’s Online Math Notes
Also you are always welcome at CAPS for drop-in tutoring, online tutoring and
learning and study groups.