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1) x = 5 2) x = 4 3) x = 15
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Exp log equations
This document discusses solving equations involving exponentials and logarithms. It provides examples of solving equations with:
1) Exponentials on both sides by taking the log of both sides.
2) Exponentials with different bases by taking the natural log of both sides.
3) Logarithms on both sides by exponentiating both sides.
4) Checking solutions for logarithmic equations to avoid extraneous solutions from properties of logs only being valid for positive inputs.
Logarithm lesson
1) The document discusses logarithms and how they were developed to simplify calculations of large numbers before calculators. Logarithms allow expressing a number as an exponent by finding what power of a base number equals the target number.
2) It provides examples of converting between exponential and logarithmic forms, using logarithmic properties to solve equations, and using change of base formulas to solve when the calculator uses a different base than the problem.
3) Key steps are shown to solve the equation 5x = 15,000 by rewriting in logarithmic form, using the change of base formula to solve for x, and checking the exponential form of the answer.
Advance algorithm hashing lec I
The document discusses hash tables as an efficient data structure for implementing dictionary operations. It describes how hash tables work by using a hash function to map keys to array indices, allowing for fast lookup times of O(1) on average. When collisions occur where two keys map to the same index, techniques like chaining can be used to resolve them. Hash tables are preferable to direct address tables when the key universe is very large relative to the actual number of keys, as they avoid wasting large amounts of memory when most array slots would be empty.
Logarithms
It includes concepts of logarithms including properties and log tables. The methodology to find out values using log tables and anti log tables is also mentioned in a detailed manner. Moreover, questions related to logarithms are mentioned for practice.
power and exponents
This document discusses scientific notation and laws of exponents that are useful for operations involving numbers with exponents. It contains 6 key points:
1) Exponents allow expressing very large and small numbers easily.
2) Numbers can be written as exponents with different bases using factorisation.
3) Laws of exponents for multiplication, division, and powers are presented with examples.
4) Scientific notation expresses numbers as a coefficient between 1-10 multiplied by a power of 10.
Advance algorithm hashing lec II
Here are the key points comparing hash-based search and binary search on a best case basis:
- Binary search has a best case time complexity of O(1) as it directly locates the target element by comparing the middle element on each iteration.
- Hash-based search has a best case time complexity of O(1) if there is no collision during probing. The target element can be found directly by indexing into the hash table.
- Collisions degrade the performance of hash-based search. The fewer collisions, the closer it gets to the best case.
- The load factor α (number of elements/number of slots) impacts the number of collisions - a lower load factor results in fewer collisions on
October 21. 2014
This document contains sample math problems involving fractions, decimals, absolute values, and equations including:
1) Converting fractions to decimals and decimals to fractions. For decimals, say the number without the decimal point to get the fraction.
2) Several absolute value equations such as 10 + 8|10 + x|= 74 and |- x - 4|= |4x + 2|.
3) An example of solving an absolute value equation by setting it equal to both positive and negative values of the absolute value.
3) logarithm definition
This document introduces logarithms and how to use them to solve exponential equations. It defines logarithms as the power to which a base number must be raised to equal the value being logged. Examples are provided of writing numbers and powers as logarithms in different bases. The basics that the logarithm of a number to itself is 1 and the logarithm of 1 or 0 is 0 or undefined are covered. Students are directed to an online worksheet and book exercises to practice evaluating and using logarithms.
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Exp log equations
This document discusses solving equations involving exponentials and logarithms. It provides examples of solving equations with:
1) Exponentials on both sides by taking the log of both sides.
2) Exponentials with different bases by taking the natural log of both sides.
3) Logarithms on both sides by exponentiating both sides.
4) Checking solutions for logarithmic equations to avoid extraneous solutions from properties of logs only being valid for positive inputs.
Logarithm lesson
1) The document discusses logarithms and how they were developed to simplify calculations of large numbers before calculators. Logarithms allow expressing a number as an exponent by finding what power of a base number equals the target number.
2) It provides examples of converting between exponential and logarithmic forms, using logarithmic properties to solve equations, and using change of base formulas to solve when the calculator uses a different base than the problem.
3) Key steps are shown to solve the equation 5x = 15,000 by rewriting in logarithmic form, using the change of base formula to solve for x, and checking the exponential form of the answer.
Advance algorithm hashing lec I
The document discusses hash tables as an efficient data structure for implementing dictionary operations. It describes how hash tables work by using a hash function to map keys to array indices, allowing for fast lookup times of O(1) on average. When collisions occur where two keys map to the same index, techniques like chaining can be used to resolve them. Hash tables are preferable to direct address tables when the key universe is very large relative to the actual number of keys, as they avoid wasting large amounts of memory when most array slots would be empty.
Logarithms
It includes concepts of logarithms including properties and log tables. The methodology to find out values using log tables and anti log tables is also mentioned in a detailed manner. Moreover, questions related to logarithms are mentioned for practice.
power and exponents
This document discusses scientific notation and laws of exponents that are useful for operations involving numbers with exponents. It contains 6 key points:
1) Exponents allow expressing very large and small numbers easily.
2) Numbers can be written as exponents with different bases using factorisation.
3) Laws of exponents for multiplication, division, and powers are presented with examples.
4) Scientific notation expresses numbers as a coefficient between 1-10 multiplied by a power of 10.
Advance algorithm hashing lec II
Here are the key points comparing hash-based search and binary search on a best case basis:
- Binary search has a best case time complexity of O(1) as it directly locates the target element by comparing the middle element on each iteration.
- Hash-based search has a best case time complexity of O(1) if there is no collision during probing. The target element can be found directly by indexing into the hash table.
- Collisions degrade the performance of hash-based search. The fewer collisions, the closer it gets to the best case.
- The load factor α (number of elements/number of slots) impacts the number of collisions - a lower load factor results in fewer collisions on
October 21. 2014
This document contains sample math problems involving fractions, decimals, absolute values, and equations including:
1) Converting fractions to decimals and decimals to fractions. For decimals, say the number without the decimal point to get the fraction.
2) Several absolute value equations such as 10 + 8|10 + x|= 74 and |- x - 4|= |4x + 2|.
3) An example of solving an absolute value equation by setting it equal to both positive and negative values of the absolute value.
3) logarithm definition
This document introduces logarithms and how to use them to solve exponential equations. It defines logarithms as the power to which a base number must be raised to equal the value being logged. Examples are provided of writing numbers and powers as logarithms in different bases. The basics that the logarithm of a number to itself is 1 and the logarithm of 1 or 0 is 0 or undefined are covered. Students are directed to an online worksheet and book exercises to practice evaluating and using logarithms.
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The document provides an assignment list that includes a test review handout due the next day, a test the following day, and a cartoon scale project due later that week. It then presents math warmup problems and lesson content on exponent rules, including the rules for multiplying and dividing terms with the same base. Examples are given to find missing exponents and simplify expressions using the exponent rules.
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The document provides instructions for a math assignment that is due the next day. It includes solving 5 warmup equations and questions 38-43 and 55-67 from lesson 3.2 of the textbook. The lesson defines linear functions as functions whose graphs are lines, and discusses solving linear functions graphically by finding the x-intercept, and algebraically by isolating the variable. It provides examples of solving linear equations both graphically and algebraically.
5 5 logarithmic functions
1. The document discusses rewriting logarithmic expressions, evaluating logarithms, and solving logarithmic equations.
2. It defines logarithms, introduces common (base 10) and natural (base e) logarithms, and explains how to "chop off the log" by rewriting logarithmic expressions in exponential form.
3. Examples are provided for evaluating logarithms and solving simple logarithmic equations.
6) equations with exponentials & logarithms
This document provides worked examples for solving various logarithmic and exponential equations. It begins with an introduction stating the learner will be able to solve exponential equations of different bases, logarithmic equations, and exponential equations of the same base. It then shows 7 worked examples of solving equations involving logarithms and exponents, with the conclusion being about expressing exponential powers in the same base when possible.
Hashing Algorithm
This document discusses dictionaries and hashing techniques for implementing dictionaries. It describes dictionaries as data structures that map keys to values. The document then discusses using a direct access table to store key-value pairs, but notes this has problems with negative keys or large memory usage. It introduces hashing to map keys to table indices using a hash function, which can cause collisions. To handle collisions, the document proposes chaining where each index is a linked list of key-value pairs. Finally, it covers common hash functions and analyzing load factor and collision probability.
Hashing
Hashing is a technique used to store and retrieve data efficiently. It involves using a hash function to map keys to integers that are used as indexes in an array. This improves searching time from O(n) to O(1) on average. However, collisions can occur when different keys map to the same index. Collision resolution techniques like chaining and open addressing are used to handle collisions. Chaining resolves collisions by linking keys together in buckets, while open addressing resolves them by probing to find the next empty index. Both approaches allow basic dictionary operations like insertion and search to be performed in O(1) average time when load factors are low.
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This document discusses evaluating and graphing logarithmic functions. It defines logarithms and explains how to evaluate logarithmic expressions without a calculator by rewriting them as exponential expressions. Examples are provided such as log 3 81 = 4 and log 2 (1/32) = -5. The definitions of common (base 10), natural (base e), and general logarithmic functions are given. Graphs of logarithmic functions are asymptotic to the horizontal axis and shift right or left depending on the base.
002#pedagogy math test
The document discusses various math concepts related to numbers and quantities for grades 3-12, including:
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3) Place value, with examples testing identification of digits in the ones, tens, hundreds, and thousands places.
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This document provides instruction on demonstrating the denseness of rational numbers using decimals. It explains that denseness means there is always a number between any two other numbers. Students are guided to write numbers vertically in a chart with added zeros and identify all numbers between given values. Examples are provided of naming numbers between decimals like 2.4 and 2.39 or converting a mixed number like 4 1/2 to a decimal. Students are assigned a handout and enrichment activity to further explain denseness through drawings.
Hashing
This document discusses hashing techniques used to store and retrieve data from hash tables. It begins by explaining why hashing is used and what a hash table is. It then describes four common hashing functions: division, multiplicative, mid square, and folded methods. The document finishes by covering techniques for resolving collisions in hash tables, including chaining, open addressing using linear probing, quadratic probing, double hashing, and rehashing.
Hash tables
The document discusses hash tables and how they can be used to implement dictionaries. Hash tables map keys to table slots using a hash function in order to store and retrieve items efficiently. Collisions may occur when multiple keys hash to the same slot. Chaining is described as a method to handle collisions by storing colliding items in linked lists attached to table slots. Analysis shows that with simple uniform hashing, dictionary operations like search, insert and delete take expected O(1) time on average.
Paper3a puertollano
This document covers topics in mathematics including:
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4 ESO Academics - Unit 03 - Exercises 4.3.4 - The Remainder Theorem. Roots of...
The document contains math exercises involving dividing polynomials using Ruffini's rule and the remainder theorem to find the remainders, identifying coefficients based on known remainders, determining if numbers are roots of polynomials, finding integer roots of polynomials, and writing polynomials with a given root. Exercises include finding remainders of dividing polynomials by (x - a), writing a remainder without dividing, finding a coefficient given a remainder, and identifying roots, integer roots, and writing polynomials with a specified root.
Hashing PPT
This document discusses hashing and different techniques for implementing dictionaries using hashing. It begins by explaining that dictionaries store elements using keys to allow for quick lookups. It then discusses different data structures that can be used, focusing on hash tables. The document explains that hashing allows for constant-time lookups on average by using a hash function to map keys to table positions. It discusses collision resolution techniques like chaining, linear probing, and double hashing to handle collisions when the hash function maps multiple keys to the same position.
Hashing Techniques in Data Structures Part2
The document discusses different approaches to handling collisions in hash tables: chaining and open addressing such as linear probing. Chaining involves storing collided keys in linked lists at each array index, while linear probing resolves collisions by probing subsequent indices in the array. The example demonstrates linear probing by inserting several keys into a hash table and showing the array indices where each key is stored.
5 5b Slope Intercept Form from Two Points
The document provides instructions for writing equations to represent lines given two points. It explains that you need the slope (m) and y-intercept (b) to write the equation in y=mx+b form. You find the slope using the rise over run formula from the two points. Then, plug the slope and the x and y values of one point into the equation to solve for b, the y-intercept. For example, given the points (2,4) and (5,7), the slope is 1, and the equation is y=x+2.
Hashing
Hashing is a technique used to store and retrieve information quickly by mapping keys to values in a hash table using a hash function. Common hash functions include division, mid-square, and folding methods. Collision resolution techniques like chaining, linear probing, quadratic probing, and double hashing are used to handle collisions in the hash table. Hashing provides constant-time lookup and is widely used in applications like databases, dictionaries, and encryption.
Data Structure and Algorithms Hashing
The document discusses hashing techniques for storing and retrieving data from memory. It covers hash functions, hash tables, open addressing techniques like linear probing and quadratic probing, and closed hashing using separate chaining. Hashing maps keys to memory addresses using a hash function to store and find data independently of the number of items. Collisions may occur and different collision resolution methods are used like open addressing that resolves collisions by probing in the table or closed hashing that uses separate chaining with linked lists. The efficiency of hashing depends on factors like load factor and average number of probes.
7) change of base
This document discusses logarithmic equations and calculations. It provides instructions on how to rewrite logarithmic equations without using logarithms, solve simultaneous logarithmic equations, calculate logarithms to specific bases and numbers of significant figures, change logarithmic bases, and solve logarithmic equations. The document aims to teach students how to perform various operations with logarithms including rewriting, solving, calculating, and changing bases.
Matrices - Discrete Structures
The document discusses matrix multiplication. Matrix multiplication involves multiplying the elements of each row of the first matrix by the elements of each column of the second matrix and adding the products. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined. The result of multiplying an m×n matrix by an n×p matrix is an m×p matrix. Matrix multiplication is not generally commutative.
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8th pre alg -l27
The document provides an assignment list that includes a test review handout due the next day, a test the following day, and a cartoon scale project due later that week. It then presents math warmup problems and lesson content on exponent rules, including the rules for multiplying and dividing terms with the same base. Examples are given to find missing exponents and simplify expressions using the exponent rules.
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The document provides instructions for a math assignment that is due the next day. It includes solving 5 warmup equations and questions 38-43 and 55-67 from lesson 3.2 of the textbook. The lesson defines linear functions as functions whose graphs are lines, and discusses solving linear functions graphically by finding the x-intercept, and algebraically by isolating the variable. It provides examples of solving linear equations both graphically and algebraically.
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1. The document discusses rewriting logarithmic expressions, evaluating logarithms, and solving logarithmic equations.
2. It defines logarithms, introduces common (base 10) and natural (base e) logarithms, and explains how to "chop off the log" by rewriting logarithmic expressions in exponential form.
3. Examples are provided for evaluating logarithms and solving simple logarithmic equations.
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This document provides worked examples for solving various logarithmic and exponential equations. It begins with an introduction stating the learner will be able to solve exponential equations of different bases, logarithmic equations, and exponential equations of the same base. It then shows 7 worked examples of solving equations involving logarithms and exponents, with the conclusion being about expressing exponential powers in the same base when possible.
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This document discusses dictionaries and hashing techniques for implementing dictionaries. It describes dictionaries as data structures that map keys to values. The document then discusses using a direct access table to store key-value pairs, but notes this has problems with negative keys or large memory usage. It introduces hashing to map keys to table indices using a hash function, which can cause collisions. To handle collisions, the document proposes chaining where each index is a linked list of key-value pairs. Finally, it covers common hash functions and analyzing load factor and collision probability.
Hashing
Hashing is a technique used to store and retrieve data efficiently. It involves using a hash function to map keys to integers that are used as indexes in an array. This improves searching time from O(n) to O(1) on average. However, collisions can occur when different keys map to the same index. Collision resolution techniques like chaining and open addressing are used to handle collisions. Chaining resolves collisions by linking keys together in buckets, while open addressing resolves them by probing to find the next empty index. Both approaches allow basic dictionary operations like insertion and search to be performed in O(1) average time when load factors are low.
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This document discusses evaluating and graphing logarithmic functions. It defines logarithms and explains how to evaluate logarithmic expressions without a calculator by rewriting them as exponential expressions. Examples are provided such as log 3 81 = 4 and log 2 (1/32) = -5. The definitions of common (base 10), natural (base e), and general logarithmic functions are given. Graphs of logarithmic functions are asymptotic to the horizontal axis and shift right or left depending on the base.
002#pedagogy math test
The document discusses various math concepts related to numbers and quantities for grades 3-12, including:
1) Prime and composite numbers, with prime numbers being only divisible by 1 and itself, and composite numbers being divisible by other numbers besides 1 and itself.
2) Prime factorization, which is expressing a number as a product of its prime factors.
3) Place value, with examples testing identification of digits in the ones, tens, hundreds, and thousands places.
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This document provides instruction on demonstrating the denseness of rational numbers using decimals. It explains that denseness means there is always a number between any two other numbers. Students are guided to write numbers vertically in a chart with added zeros and identify all numbers between given values. Examples are provided of naming numbers between decimals like 2.4 and 2.39 or converting a mixed number like 4 1/2 to a decimal. Students are assigned a handout and enrichment activity to further explain denseness through drawings.
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This document discusses hashing techniques used to store and retrieve data from hash tables. It begins by explaining why hashing is used and what a hash table is. It then describes four common hashing functions: division, multiplicative, mid square, and folded methods. The document finishes by covering techniques for resolving collisions in hash tables, including chaining, open addressing using linear probing, quadratic probing, double hashing, and rehashing.
Hash tables
The document discusses hash tables and how they can be used to implement dictionaries. Hash tables map keys to table slots using a hash function in order to store and retrieve items efficiently. Collisions may occur when multiple keys hash to the same slot. Chaining is described as a method to handle collisions by storing colliding items in linked lists attached to table slots. Analysis shows that with simple uniform hashing, dictionary operations like search, insert and delete take expected O(1) time on average.
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This document covers topics in mathematics including:
1. Linear algebra concepts like partitioned matrices and block multiplication
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4. Evaluating integrals using trigonometric substitutions
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The document contains math exercises involving dividing polynomials using Ruffini's rule and the remainder theorem to find the remainders, identifying coefficients based on known remainders, determining if numbers are roots of polynomials, finding integer roots of polynomials, and writing polynomials with a given root. Exercises include finding remainders of dividing polynomials by (x - a), writing a remainder without dividing, finding a coefficient given a remainder, and identifying roots, integer roots, and writing polynomials with a specified root.
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This document discusses hashing and different techniques for implementing dictionaries using hashing. It begins by explaining that dictionaries store elements using keys to allow for quick lookups. It then discusses different data structures that can be used, focusing on hash tables. The document explains that hashing allows for constant-time lookups on average by using a hash function to map keys to table positions. It discusses collision resolution techniques like chaining, linear probing, and double hashing to handle collisions when the hash function maps multiple keys to the same position.
Hashing Techniques in Data Structures Part2
The document discusses different approaches to handling collisions in hash tables: chaining and open addressing such as linear probing. Chaining involves storing collided keys in linked lists at each array index, while linear probing resolves collisions by probing subsequent indices in the array. The example demonstrates linear probing by inserting several keys into a hash table and showing the array indices where each key is stored.
5 5b Slope Intercept Form from Two Points
The document provides instructions for writing equations to represent lines given two points. It explains that you need the slope (m) and y-intercept (b) to write the equation in y=mx+b form. You find the slope using the rise over run formula from the two points. Then, plug the slope and the x and y values of one point into the equation to solve for b, the y-intercept. For example, given the points (2,4) and (5,7), the slope is 1, and the equation is y=x+2.
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Hashing is a technique used to store and retrieve information quickly by mapping keys to values in a hash table using a hash function. Common hash functions include division, mid-square, and folding methods. Collision resolution techniques like chaining, linear probing, quadratic probing, and double hashing are used to handle collisions in the hash table. Hashing provides constant-time lookup and is widely used in applications like databases, dictionaries, and encryption.
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The document discusses hashing techniques for storing and retrieving data from memory. It covers hash functions, hash tables, open addressing techniques like linear probing and quadratic probing, and closed hashing using separate chaining. Hashing maps keys to memory addresses using a hash function to store and find data independently of the number of items. Collisions may occur and different collision resolution methods are used like open addressing that resolves collisions by probing in the table or closed hashing that uses separate chaining with linked lists. The efficiency of hashing depends on factors like load factor and average number of probes.
7) change of base
This document discusses logarithmic equations and calculations. It provides instructions on how to rewrite logarithmic equations without using logarithms, solve simultaneous logarithmic equations, calculate logarithms to specific bases and numbers of significant figures, change logarithmic bases, and solve logarithmic equations. The document aims to teach students how to perform various operations with logarithms including rewriting, solving, calculating, and changing bases.
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Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Day 4 examples u6f13
- 1. Mental Math Solve for x 1) 32 3x - 2 2) = 16 1 = 64 2x-1 3) 3(52x - 9) = 375 4) log81( 1 ) = x 3 5) log8x = 3 2
- 2. Intro to Using Logs We'll use log = log button on calculator, called 'common log' Can't rewrite as same base Instead, we can take or apply logs to both sides and solve = log15 Solve for x Can NEVER rewrite as a power of 4 Can only solve by taking logs of both sides 2) Expand the logs 3) Solve for x