Understanding the Language of Algebra and Mathematical Concepts
Basic rules in both language and mathematics are crucial for success. By grasping key concepts such as number types and classifications, functions, transformations, and more, students can enhance their logical thinking and problem-solving skills. This refresher introduces essential math topics like natural numbers, integers, rational numbers, irrational numbers, and real numbers, providing a foundation for tackling algebraic challenges with confidence.
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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 Number Types Number Types Functions Functions Transformations Transformations Factoring Factoring Distributing Distributing Fractions Fractions Exponents Exponents Solving Quadratic Equations Solving Quadratic Equations Logarithms and Exponentials Logarithms and Exponentials
NUMBER TYPES AND CLASSIFICATIONS Natural Numbers 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 or a boldface N. N. Integers 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 or a boldface Z. Z. Rational Numbers Pick any two integers from , a and b. Then any fraction ? b 0 is a rational number. For instance then, 7 numbers. They are denoted as or a boldface Q. Rational Numbers ?for 3 89,2 1,0 etc. are all rational 11, Q.
NUMBER TYPES AND CLASSIFICATIONS Irrational Numbers 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 ?. 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?) Real Numbers 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 or a boldface R. 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 N belongs to Z Z which belongs to Q Q which belongs to R. R.)
NUMBER TYPES AND CLASSIFICATION Imaginary Numbers 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 = ? ??? 9 = 3?. . Often times a problem will ask you to find all real solutions . What does this mean? ? 32= 49 ? 3 = ? = 7? + 3 49 = 7? 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 or a boldface C. C.
NUMBER TYPES AND CLASSIFICATIONS Here are some problems for you to work out on your own: Classify each number: Classify each number: . ? 1. 1. 2. 2. ? 3. 3. ??.? 4. 4. ?? 5. 5. ?,???? ?? 6. 6. 7. 7. ? Answers: 1) rational 2) irrational 3) rational 4) imaginary 5) imaginary 6) natural 7) integer Answers: 1) rational 2) irrational 3) rational 4) imaginary 5) imaginary 6) natural 7) integer
FUNCTIONS What is a function? What is a function? Mathematically: 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: 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 Functions can be defined in lots of different ways. Just about every function we see would look something like this: ? ? = 2? + 1 The name of our function is ? 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. 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 : 1 ? or g ? = ?. ? ? = Why do we have domain problems here?
FUNCTIONS Example: Find the domain of the following function Example: Find the domain of the following function ? ? =2? 6 8? 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. 8? 4 = 0 8? = 4 ? =1 2 Therefore we would write our domain as: ?: ,1 1 2, 2
FUNCTIONS Example: Example: 2? ? ? = ? 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. ? 8 > 0 ? > 8 Our domain is then: ?: 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 ? ? ?? ? values. For instance, let us think of the graph of ? ? = ?. 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: Find the domain of the following functions: 1. 1. ? ? = ?? ? 2. 2. ? ? = ?? ?? ?? ?? 4. 4. ? ? = ?? ? 3. 3. ? ? = ? 5. 5. ? ? = ? ? 3) 3) ,?? ?? ?, ? ?, Answers: 1) Answers: 1) , 2) 2) ( , ) 4) 4) 5) 5) [?,?) ?, ?
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 ? ? then the transformed function ?(?) will look something like this: ? ? = ?? ?? + ? + ? 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 ? ? = ?? ?? + ? + ? Inside f(x): Step 1 Step 1 The c term will give the horizontal shift. Step 2 Step 2 If there is a - sign then we will have a horizontal reflection about the y-axis. Step 3 Step 3 The b term gives a horizontal stretch/compression. Outside f(x): Step 4 Step 4 The - will give us a vertical reflection about the x axis. Step 5 Step 5 - The a term will produce a vertical stretch/compression. Step 6 Step 6 The d term will give rise to a vertical shift.
TRANSFORMATIONS Example: Example: Sketch the graph of ? ? = ? 2 + 4 . Here our base function is ?. This graph looks like:
TRANSFORMATIONS We next look for what s inside our base function: That will be ? ? = This will result in a shift 2 units to the right. ? 2 (green).
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. ? ? = ? 2. (orange)
TRANSFORMATIONS Finally we account for the vertical shift. ? ? = ? 2 + 4(black)
TRANSFORMATIONS The domain and range of the base function ? ? = ?: ?: 0, ?:[0, ) The domain and range of our new function ? ? = ? 2 + 4 ?:[2, ) ?:( ,4] How could we find the domain algebraically?
FACTORING Factors Factors two numbers or terms that can be multiplied together to get another two numbers or terms that can be multiplied together to get another term or number. term or number. 2 and 3 are factors of 6. ? 3 ??? 2? 6 are factors of 2?2 12? + 18. (Is that clear as to why?) 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 ? ? = 3?3 27?2+ 27?
FACTORING ? ? = 3?3 27?2+ 27? First Step: 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. ? ? = 3 ?3 9?2+ 9? Second Step: 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. ? ? = 3?(?2 9? + 9)
FACTORING Here are some problems for you to work out on your own. Factor the following functions: Factor the following functions: ???+ ???+ ?? 1. 1. ?? ? ?? 2. 2. ?? ? ? ? + ? ? (? ?) 3. 3. Answers: 1) Answers: 1) ??(???+ ? + ?) 3) 3) ?? ? ? ??+ ? ? 2) 2) (? ?)(? + ?)
DISTRIBUTION Distribution is just reverse factoring. It undoes what we factored. 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. ? ? = 2?2? 4 ?3 9 = 2?3 8?2 ?3 9 = 2?6 18?3 8?5+ 72?2
DISTRIBUTION Example: Distribute he following function: Example: Distribute he following function: ? ? = 3? ? 6 ? 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. ? ? = 3? ? 6 ? 4 ? ? = 3? ?2 4? 6? + 24 ? ? = 3? ?2 10? + 24 ? ? = 3?3 30?2+ 72?
DISTRIBUTION Here are some problems for you to work out on your own. Distribute the following expressions: Distribute the following expressions: ?? ?? ? ? 1. 1. ? ? ? ? + ? ? ? 2. 2. ? ??? ??? ? 3. 3. ??? ? 4. 4. ? ? ?? ? ?+ ?? ?? ? ?? ??? Answers: 1) Answers: 1) ??+ ??? ??? 3) 3) ??? 4) 4) ?? ??? 2) 2) ?
FRACTIONS 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. ?2+ 4 ? 2+ 3?? 9 =?2+ 4 =?2+ 4 =7 ?2+ 4 + (? 2)(3?2 27?) 7(? 2) =7?2+ 28 + 3?3 27?2 6?2+ 54? 7? 14 7 ? 2+3?(? 9) ? 2+3?2 27? 7 7 =3?3 26?2+ 54? + 28 7? 14
FRACTIONS Here are some problems for you to work out on you own: Simplify the following fractions: Simplify the following fractions: ?? ? ? ?+??+? ? ? ? ? ?? ???+?+ 1. 1. ? ? 2. 2. ? 3. 3. ??? ? Answers: 1) ??? 2) ?? ?? 3) ????+??? ??+? ???? ? Answers: 1) 2) 3) ? ? ??
EXPONENTS 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): ????= ??+? ?? ??= ?? ? ???= ???? ? =?? ? ? ?? ?? ?= ???
EXPONENTS Example: Simplify the following expression: Example: Simplify the following expression: 27?7??4 3???8 1 2 1?4 8 = 9?7 1? = 9?6? 1 2? 4 9?6 ??4 =
EXPONENTS Here are some problems for you to work out on your own. Example: Simplify the following expressions: Example: Simplify the following expressions: ????? ?? 2. 2. ????? ????? ????? 1. 1. ?????? 3. 3. ? ? 3) 3) ? Answers: 1) Answers: 1) ???? 2) 2) ? ???
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 formula. 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. general factoring and the second is through the quadratic quadratic 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. ?2+ ? 12 = 0
SOLVING QUADRATIC EQUATIONS First Way: Factoring First Way: Factoring ?2+ ? 12 = 0 = ? 3 ? + 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. ? 3 = 0 ? + 4 = 0 ? = 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 Second Way: Quadratic Formula Second Way: Quadratic Formula If we have the quadratic equation in the standard form: ??2+ ?? + ? = 0 Then the quadratic formula solves for x (every time!): ? = ? ?2 4?? 2?
SOLVING QUADRATIC EQUATIONS Our problem was: ?2+ ? 12 = 0 Because we wish to use the quadratic formula we need to find a, b, and c. ? = 1,? = 1,? = 12 Plugging this into the quadratic equation yields: ? = ? ?2 4?? 12 (4)(1)( 12) 2(1) = 1 7 2 = 1 2? ? = 1 49 2 Again, we have two equations to solve: ? = 1 + 7 ??? ? = 1 7 2 2
SOLVING QUADRATIC EQUATIONS 1 + 7 2 ? =6 ??? ? = 1 7 2??? ? = 8 ? = 2 2 ? = 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: Example: Solve the following quadratic equations: ??+ ? ? ?? ?? + ? ???+ ?? + ? ??+ ?? ? 1. 1. 2. 2. 3. 3. 4. 4. 3) 3) ? = ? ? ? Answers: 1) Answers: 1) ? = ?, ? 2) 2) ? = ?,? 4) 4) ? = ?, ? ?
LOGS AND EXPONENTIALS An exponential function is any function of the form: ? ? = ??,? ??? ? ?? ? ?????? A logarithm is the inverse inverse of the exponential. What this means is that if we have an exponential ??= ? log?? = ?. The most common exponential is ??. The inverse for the exponential : ??= ? ln? = ? 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!) remember they re inverses!)
LOGS AND EXPONENTS Exponential Functions: ?: , ?: 0, Logarithmic Functions: ?: 0, ?: , Let s now use our transformation rules for the graph of ? ? = ??+ 2
LOGS AND EXPONENTS Our base function (red) is ? ? = ??. 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 ? ? = ??+ 2:
LOGS AND EXPONENTS Our new domain and range will be: ?: , ?:( ,2) How can we find our x and y intercepts algebraically now?
RECAP 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.
