Functions O level
•Download as PPSX, PDF•
0 likes•73 views
This document provides examples of how to evaluate different types of algebraic formulas and functions. It contains examples of evaluating a formula for specific given values, finding the value of a function for a given input, and solving simultaneous equations to determine the coefficients of a linear function.
Report
Share
Report
Share
Recommended
Vector space introduction
Vectors can represent physical quantities that have both magnitude and direction. They are represented by arrows with length and direction. Scalar multiplication multiplies the magnitude of a vector by a real number while keeping or reversing the direction. Vectors in n-dimensional space are n-tuples of real numbers. Vector addition combines the corresponding components, while scalar multiplication multiplies each component by the scalar. Matrices are arrays of numbers used to represent linear systems of equations.
Definition Vector space
This document introduces the definition of a vector space, which consists of a set V with two operations: vector addition and scalar multiplication. For a set V to be a vector space over a field K, it must satisfy eight axioms relating to vector addition, scalar multiplication, the existence of a zero vector, and the existence of negatives. Examples of vector spaces include the set of all n-tuples over a field K under component-wise addition and scalar multiplication.
Prerequisite for metric space
The document defines sets, elements, and operations on sets. It discusses that a set is a collection of objects, called elements or points. It provides examples of sets and defines subset and subset notation. The document then defines functions, including the domain, codomain, range, and image of a function. It discusses inverse functions and inverse image notation. Finally, it discusses characteristics functions, where the characteristics function of a set A maps elements in A to 1 and elements not in A to 0, completely describing the set.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Class 5 - Set Theory and Venn Diagrams
This document covers set theory, Venn diagrams, and their applications. It introduces key concepts in set theory, such as sets, elements, subsets, unions, intersections, complements and cardinality. It also explains how to represent relationships between sets visually using Venn diagrams and how to solve problems using Venn diagrams. Examples are provided for set operations and using Venn diagrams to calculate values for skills problems. Students are assigned exercises to practice these set theory and Venn diagram concepts.
Evaluating Expressions Review
The document provides 12 math problems that involve evaluating expressions when given values for x and y or evaluating expressions when given a value for y. The expressions include addition, subtraction, multiplication, division, exponents, and variables. Students are asked to calculate the numerical value of each expression when provided the values for the variables.
Chapter i
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
Methods of writing sets
There are two methods to represent or write a set:
1. Listing Method/Roster Method: All elements of a set are listed within curly brackets, with each element written only once and separated by commas.
2. Rule Method/Set Builder Form: Instead of listing elements, a variable is used followed by a property that all elements of the set possess. This property describes the common traits of all elements in the set.
Recommended
Vector space introduction
Vectors can represent physical quantities that have both magnitude and direction. They are represented by arrows with length and direction. Scalar multiplication multiplies the magnitude of a vector by a real number while keeping or reversing the direction. Vectors in n-dimensional space are n-tuples of real numbers. Vector addition combines the corresponding components, while scalar multiplication multiplies each component by the scalar. Matrices are arrays of numbers used to represent linear systems of equations.
Definition Vector space
This document introduces the definition of a vector space, which consists of a set V with two operations: vector addition and scalar multiplication. For a set V to be a vector space over a field K, it must satisfy eight axioms relating to vector addition, scalar multiplication, the existence of a zero vector, and the existence of negatives. Examples of vector spaces include the set of all n-tuples over a field K under component-wise addition and scalar multiplication.
Prerequisite for metric space
The document defines sets, elements, and operations on sets. It discusses that a set is a collection of objects, called elements or points. It provides examples of sets and defines subset and subset notation. The document then defines functions, including the domain, codomain, range, and image of a function. It discusses inverse functions and inverse image notation. Finally, it discusses characteristics functions, where the characteristics function of a set A maps elements in A to 1 and elements not in A to 0, completely describing the set.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Class 5 - Set Theory and Venn Diagrams
This document covers set theory, Venn diagrams, and their applications. It introduces key concepts in set theory, such as sets, elements, subsets, unions, intersections, complements and cardinality. It also explains how to represent relationships between sets visually using Venn diagrams and how to solve problems using Venn diagrams. Examples are provided for set operations and using Venn diagrams to calculate values for skills problems. Students are assigned exercises to practice these set theory and Venn diagram concepts.
Evaluating Expressions Review
The document provides 12 math problems that involve evaluating expressions when given values for x and y or evaluating expressions when given a value for y. The expressions include addition, subtraction, multiplication, division, exponents, and variables. Students are asked to calculate the numerical value of each expression when provided the values for the variables.
Chapter i
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
Methods of writing sets
There are two methods to represent or write a set:
1. Listing Method/Roster Method: All elements of a set are listed within curly brackets, with each element written only once and separated by commas.
2. Rule Method/Set Builder Form: Instead of listing elements, a variable is used followed by a property that all elements of the set possess. This property describes the common traits of all elements in the set.
Integral calculus
1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a
Rules of writing sets
(1) A set is defined by capital letters and contains elements written within curly brackets. (2) Elements of a set are represented by small letters and the symbol ∈ is used to indicate that an element belongs to a set. (3) The document provides examples of sets A and D, where A = {1,2,3,4,5,6,7,8,9,10} and D = {2, 3, 5, 7, 11, 13}. It indicates which elements belong to each set using the ∈ symbol.
Evaluating functions basic rules
This document provides an overview of evaluating functions by giving examples of evaluating functions for given values and providing function rules to evaluate for specific inputs. It emphasizes important rules like exponents, adding and subtracting integers, and order of operations when evaluating functions and provides additional practice problems for readers to evaluate functions based on the given rules.
Relations and Functions 2
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
Eighan values and diagonalization
This document discusses eigenvalues and diagonalization of matrices. Some key points:
- Eigenvalues are values that satisfy the equation AX = λX, where X is a non-zero eigenvector. The eigenspace associated with λ contains all eigenvectors for that eigenvalue.
- The characteristic polynomial of a matrix A is defined as det(xI - A). The eigenvalues of A are the roots of its characteristic polynomial.
- Similar matrices have the same eigenvalues, determinant, rank, trace, and characteristic polynomial. Two matrices are similar if one can be obtained from the other by conjugation via an invertible matrix.
- The trace of a matrix is the sum of its diagonal entries
9.4 part 4.ppt worked
1) A geometric series is the expression of the sum of terms in a geometric sequence. The sum of a finite geometric series can be calculated using the formula Sn = a(1 - r^n)/(1 - r) where a is the first term, r is the common ratio, and n is the number of terms.
2) Geometric series converge when the absolute value of the common ratio r is less than 1, and diverge when the absolute value of r is greater than 1.
3) The homework assignment includes evaluating example geometric series, determining whether other series converge or diverge, and completing a worksheet to review for a test on geometric series.
Arithmetic progressions - Poblem based Arithmetic progressions
Arithmetic progressions - problem based Arithmetic progressions.
Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring.
Our Mission- Our aspiration is to be a renowned unpaid school on Web-World.
Contact Us -
Website - www.letstute.com
YouTube - www.youtube.com/letstute
Linear functions
This document defines and compares linear equations and linear functions. It explains that a linear equation involves one variable with the highest power being 1, while a linear function relates two variables with the highest power of one variable being 1. The document also defines domain as the set of input values and range as the set of output values. An example shows how to identify the domain and range from a set of ordered pairs for a linear function.
Metric space
This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.
Database systems-Formal relational query languages
The document discusses relational algebra and tuple relational calculus. It defines the basic operators of relational algebra including selection, projection, union, difference, Cartesian product, and rename. It provides examples of how to write queries using each operator. It also discusses tuple relational calculus, defining domains, predicates, quantifiers, and how to write safe queries using this calculus.
Intervensi Modul 1 Simultanious
This document provides 15 problems involving solving simultaneous linear equations. Each problem has 4 marks allocated and requires calculating the values of variables (like m and n) that satisfy two given simultaneous linear equations. The document then provides the step-by-step working and solutions for each of the 15 problems.
Crisp set
The document discusses classical or crisp set theory. Some key points:
1) Classical set theory deals with sets that have definite membership - an element either fully belongs to a set or not. This is represented by true/false or yes/no.
2) A set is a well-defined collection of objects. The universal set is the overall context within which sets are defined.
3) Set operations like union, intersection, complement and difference are used to combine or relate sets according to specific rules.
4) Properties like commutativity, associativity and distributivity define the logical behavior of sets under different operations.
sets by navneet
This document contains information about sets and set operations. It defines what a set is, provides examples of standard sets like natural numbers and real numbers, and discusses concepts like subsets, power sets, Cartesian products, unions, intersections, differences and complements of sets. It also presents methods to prove equations involving set operations.
Lecture 11 diagonalization & complex eigenvalues - 5-3 & 5-5
This document discusses diagonalization of matrices. It defines similarity of matrices and notes that similar matrices have the same characteristic polynomial and eigenvalues. It then discusses diagonalizing matrices by finding the eigenvalues and corresponding eigenvectors, constructing a change of basis matrix P from the eigenvectors, and constructing a diagonal matrix D from the eigenvalues. It provides examples of diagonalizing matrices with real and complex eigenvalues.
Aaprogresiones -aritmeticas_(2)
The document discusses arithmetic progressions and provides formulas for:
1) The nth term (an) of an arithmetic progression as an = a1 + (n-1)d, where a1 is the first term and d is the common difference.
2) The sum (Sn) of the first n terms as Sn = (n/2)(a1 + an), where a1 is the first term and an is the nth term.
3) It provides examples of using these formulas to calculate the nth term, number of terms, and sum of terms for various arithmetic progressions.
Lesson 7 antidifferentiation generalized power formula-simple substitution
This document discusses antiderivatives and indefinite integration. It defines an antiderivative as a function whose derivative is the given function. The notation for antiderivatives uses an integral sign to indicate finding a function with the given differential. Several properties of indefinite integrals are described, including the power rule and that the integral of a sum is the sum of the integrals. Examples of integrating various functions using properties like substitution are provided.
Open sentences.ppt
The document defines key terms related to expressions, equations, and inequalities. It provides examples of each and has students identify whether examples given are expressions, equations, or inequalities. It also covers solving equations by applying order of operations and defining variables, replacement sets, and solution sets when solving equations and inequalities.
Multimedia project
The math tutoring session covered solving equations using order of operations and variables, calculating perimeter, area, and circumference, and word problems involving rates and time. Key objectives were for the student to learn order of operations, solve equations, and review calculating perimeter, area, and circumference. Sample problems were provided to demonstrate solving techniques.
Quarter 1 - Illustrating and solving quadratic equations
1) Quadratic equations are mathematical expressions of degree 2 that can be written in the standard form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
2) There are four main methods for solving quadratic equations: extracting square roots, factoring, completing the square, and using the quadratic formula.
3) Factoring involves transforming the quadratic expression into standard form, factoring the expression into two linear factors, and setting each factor equal to 0 to solve for the roots.
1.2 simplifying expressions and order of operations
1. The document discusses simplifying algebraic expressions by identifying parts such as terms, coefficients, and constants. It also covers combining like terms.
2. It explains the order of operations as PEMDAS and uses it to evaluate expressions. Parentheses have the highest priority, then exponents, multiplication, division, addition and subtraction from left to right.
3. Formulas are mathematical relationships between quantities that can be used to find unknown values. Examples of using formulas to find the area of geometric shapes are provided.
Gen Math topic 1.pptx
This document provides an overview of functions and their key concepts. It defines relations, domains, ranges, and functions. It discusses different types of functions including constant, linear, quadratic, cubic, and others. It also covers evaluating functions, performing operations on functions, and piecewise functions. The document is intended to help understand functions and how they can represent real-life situations. It provides examples of evaluating functions at different inputs, adding and subtracting functions, and finding function values for piecewise functions.
Unit-1 Basic Concept of Algorithm.pptx
The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.
More Related Content
What's hot
Integral calculus
1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a
Rules of writing sets
(1) A set is defined by capital letters and contains elements written within curly brackets. (2) Elements of a set are represented by small letters and the symbol ∈ is used to indicate that an element belongs to a set. (3) The document provides examples of sets A and D, where A = {1,2,3,4,5,6,7,8,9,10} and D = {2, 3, 5, 7, 11, 13}. It indicates which elements belong to each set using the ∈ symbol.
Evaluating functions basic rules
This document provides an overview of evaluating functions by giving examples of evaluating functions for given values and providing function rules to evaluate for specific inputs. It emphasizes important rules like exponents, adding and subtracting integers, and order of operations when evaluating functions and provides additional practice problems for readers to evaluate functions based on the given rules.
Relations and Functions 2
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
Eighan values and diagonalization
This document discusses eigenvalues and diagonalization of matrices. Some key points:
- Eigenvalues are values that satisfy the equation AX = λX, where X is a non-zero eigenvector. The eigenspace associated with λ contains all eigenvectors for that eigenvalue.
- The characteristic polynomial of a matrix A is defined as det(xI - A). The eigenvalues of A are the roots of its characteristic polynomial.
- Similar matrices have the same eigenvalues, determinant, rank, trace, and characteristic polynomial. Two matrices are similar if one can be obtained from the other by conjugation via an invertible matrix.
- The trace of a matrix is the sum of its diagonal entries
9.4 part 4.ppt worked
1) A geometric series is the expression of the sum of terms in a geometric sequence. The sum of a finite geometric series can be calculated using the formula Sn = a(1 - r^n)/(1 - r) where a is the first term, r is the common ratio, and n is the number of terms.
2) Geometric series converge when the absolute value of the common ratio r is less than 1, and diverge when the absolute value of r is greater than 1.
3) The homework assignment includes evaluating example geometric series, determining whether other series converge or diverge, and completing a worksheet to review for a test on geometric series.
Arithmetic progressions - Poblem based Arithmetic progressions
Arithmetic progressions - problem based Arithmetic progressions.
Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring.
Our Mission- Our aspiration is to be a renowned unpaid school on Web-World.
Contact Us -
Website - www.letstute.com
YouTube - www.youtube.com/letstute
Linear functions
This document defines and compares linear equations and linear functions. It explains that a linear equation involves one variable with the highest power being 1, while a linear function relates two variables with the highest power of one variable being 1. The document also defines domain as the set of input values and range as the set of output values. An example shows how to identify the domain and range from a set of ordered pairs for a linear function.
Metric space
This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.
Database systems-Formal relational query languages
The document discusses relational algebra and tuple relational calculus. It defines the basic operators of relational algebra including selection, projection, union, difference, Cartesian product, and rename. It provides examples of how to write queries using each operator. It also discusses tuple relational calculus, defining domains, predicates, quantifiers, and how to write safe queries using this calculus.
Intervensi Modul 1 Simultanious
This document provides 15 problems involving solving simultaneous linear equations. Each problem has 4 marks allocated and requires calculating the values of variables (like m and n) that satisfy two given simultaneous linear equations. The document then provides the step-by-step working and solutions for each of the 15 problems.
Crisp set
The document discusses classical or crisp set theory. Some key points:
1) Classical set theory deals with sets that have definite membership - an element either fully belongs to a set or not. This is represented by true/false or yes/no.
2) A set is a well-defined collection of objects. The universal set is the overall context within which sets are defined.
3) Set operations like union, intersection, complement and difference are used to combine or relate sets according to specific rules.
4) Properties like commutativity, associativity and distributivity define the logical behavior of sets under different operations.
sets by navneet
This document contains information about sets and set operations. It defines what a set is, provides examples of standard sets like natural numbers and real numbers, and discusses concepts like subsets, power sets, Cartesian products, unions, intersections, differences and complements of sets. It also presents methods to prove equations involving set operations.
Lecture 11 diagonalization & complex eigenvalues - 5-3 & 5-5
This document discusses diagonalization of matrices. It defines similarity of matrices and notes that similar matrices have the same characteristic polynomial and eigenvalues. It then discusses diagonalizing matrices by finding the eigenvalues and corresponding eigenvectors, constructing a change of basis matrix P from the eigenvectors, and constructing a diagonal matrix D from the eigenvalues. It provides examples of diagonalizing matrices with real and complex eigenvalues.
Aaprogresiones -aritmeticas_(2)
The document discusses arithmetic progressions and provides formulas for:
1) The nth term (an) of an arithmetic progression as an = a1 + (n-1)d, where a1 is the first term and d is the common difference.
2) The sum (Sn) of the first n terms as Sn = (n/2)(a1 + an), where a1 is the first term and an is the nth term.
3) It provides examples of using these formulas to calculate the nth term, number of terms, and sum of terms for various arithmetic progressions.
Lesson 7 antidifferentiation generalized power formula-simple substitution
This document discusses antiderivatives and indefinite integration. It defines an antiderivative as a function whose derivative is the given function. The notation for antiderivatives uses an integral sign to indicate finding a function with the given differential. Several properties of indefinite integrals are described, including the power rule and that the integral of a sum is the sum of the integrals. Examples of integrating various functions using properties like substitution are provided.
What's hot (16)
Arithmetic progressions - Poblem based Arithmetic progressions
Arithmetic progressions - Poblem based Arithmetic progressions
Database systems-Formal relational query languages
Database systems-Formal relational query languages
Lecture 11 diagonalization & complex eigenvalues - 5-3 & 5-5
Lecture 11 diagonalization & complex eigenvalues - 5-3 & 5-5
Lesson 7 antidifferentiation generalized power formula-simple substitution
Lesson 7 antidifferentiation generalized power formula-simple substitution
Similar to Functions O level
Open sentences.ppt
The document defines key terms related to expressions, equations, and inequalities. It provides examples of each and has students identify whether examples given are expressions, equations, or inequalities. It also covers solving equations by applying order of operations and defining variables, replacement sets, and solution sets when solving equations and inequalities.
Multimedia project
The math tutoring session covered solving equations using order of operations and variables, calculating perimeter, area, and circumference, and word problems involving rates and time. Key objectives were for the student to learn order of operations, solve equations, and review calculating perimeter, area, and circumference. Sample problems were provided to demonstrate solving techniques.
Quarter 1 - Illustrating and solving quadratic equations
1) Quadratic equations are mathematical expressions of degree 2 that can be written in the standard form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
2) There are four main methods for solving quadratic equations: extracting square roots, factoring, completing the square, and using the quadratic formula.
3) Factoring involves transforming the quadratic expression into standard form, factoring the expression into two linear factors, and setting each factor equal to 0 to solve for the roots.
1.2 simplifying expressions and order of operations
1. The document discusses simplifying algebraic expressions by identifying parts such as terms, coefficients, and constants. It also covers combining like terms.
2. It explains the order of operations as PEMDAS and uses it to evaluate expressions. Parentheses have the highest priority, then exponents, multiplication, division, addition and subtraction from left to right.
3. Formulas are mathematical relationships between quantities that can be used to find unknown values. Examples of using formulas to find the area of geometric shapes are provided.
Gen Math topic 1.pptx
This document provides an overview of functions and their key concepts. It defines relations, domains, ranges, and functions. It discusses different types of functions including constant, linear, quadratic, cubic, and others. It also covers evaluating functions, performing operations on functions, and piecewise functions. The document is intended to help understand functions and how they can represent real-life situations. It provides examples of evaluating functions at different inputs, adding and subtracting functions, and finding function values for piecewise functions.
Unit-1 Basic Concept of Algorithm.pptx
The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.
Presentation2.pptx
Evaluating a function means replacing a variable in the function with a value from the domain and computing the result. For example, to evaluate f(x) = 2x + 4 for x = 5, replace x with 5: f(5) = 2×5 + 4 = 14. It is important to see that the variable is just a placeholder and the name of the function is arbitrary. Evaluating a function can mean replacing the variable with a number or an expression.
Expressions and variables
The document defines numerical expressions, variables, and variable expressions. It provides examples of how to evaluate variable expressions by substituting number values for the variables and evaluating the resulting numerical expression. Four examples are shown with step-by-step solutions for evaluating expressions containing multiple variables when given the values of the variables.
REPRESENTATION OF FUNCTIONS.pptx
This document discusses functions and their representations. It begins by defining relations and functions, and providing examples of each. It then discusses different types of functions including linear, quadratic, constant, identity, absolute value, and piecewise functions. Examples are provided for each type. The document ends with exercises asking the reader to determine if given relations are functions, and to identify what type of function is being described.
Evaluating Algebraic Expressions
Algebra involves the study of variables represented by letters and the operations relating these variables. The document discusses evaluating algebraic expressions by substituting numbers for variables and simplifying. Examples are provided of evaluating expressions when given values for variables, along with drills for readers to practice evaluating expressions using specified variable values.
Quadratic Functions.pptx
This document provides information about quadratic functions including:
1. Quadratic functions have graphs that form a U-shaped curve called a parabola with a single turning point called the vertex.
2. The axis of symmetry of the parabola passes through the vertex and divides the graph into two symmetrical parts.
3. Quadratic functions can be transformed between the standard form f(x)=ax^2 + bx + c and the vertex form f(x)=a(x-h)^2 + k using completing the square. This allows finding the vertex (h,k) of the parabola.
4. Examples show how to transform quadratic functions between the standard and vertex
Fst ch3 notes
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
BASIC OF ALGORITHM AND MATHEMATICS STUDENTS
An algorithm is a step-by-step procedure to solve problems. It must be unambiguous, finite, and precisely defined. The document discusses types of algorithms and methods for multiplication. It also covers basics of set theory including sets, subsets, operations, relations, and equivalence relations. Key concepts like functions, vectors, and matrices are introduced for understanding algorithms and mathematics.
Laplace Transform and its applications
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Module III - 1 - Orders of Growth.p df
The document discusses algorithm analysis and asymptotic notation. It introduces orders of growth, best-case, worst-case and average-case efficiencies. The key asymptotic notations of big-O, big-Omega and big-Theta are defined and examples are provided to illustrate how to determine the asymptotic notation of algorithms. The analysis of both recursive and non-recursive algorithms is covered, along with examples such as sequential search and finding the maximum element in an array.
Lesson 21: More Algebra
This document provides a lesson summary for topics in algebra including:
1) Operations on functions such as sum, difference, product, and quotient of functions. It also covers function composition and inverse functions.
2) Solving quadratic equations through factoring and using the quadratic formula.
3) Graphs of polynomial functions and applying the factor theorem, remainder theorem, and rational zero theorem to polynomials.
4) Determining the domains of functions when combining, composing, or taking inverses of functions.
The document includes examples and step-by-step solutions for applying these algebraic concepts.
01 FUNCTIONS.pptx
This powerpoint presentation gives information regarding functions. Designed or grade 11 studens studying general mathematics 11. You can use this presentation to present your lessons in grade 11 general mathematics or even use this on your lesson in grade 10 mathematics about polynomial functions
Chapter1.1
The document provides examples and explanations for evaluating algebraic expressions. It introduces key vocabulary like expression, variable, numerical expression, algebraic expression, and evaluate. Examples are provided for evaluating expressions with one variable, two variables, and applications involving converting between Celsius and Fahrenheit temperatures. Practice problems are included at the end to evaluate expressions for given variable values.
Lecture 3 (algebraic expressions)
This document provides information and examples about algebraic expressions and evaluating expressions using order of operations. It includes:
- Definitions of variables, expressions, operations like addition and multiplication.
- Examples of writing expressions for word phrases and evaluating numerical expressions using order of operations rules like BODMAS/PEMDAS.
- Questions asking to write expressions from word phrases, evaluate expressions for given variable values, and evaluating variable expressions.
The document covers basic topics of writing and evaluating algebraic expressions and the order in which operations should be performed. It provides examples to illustrate definitions and multiple practice questions.
Module For Mathematics
The document provides information about algebraic expressions and polynomials. It begins by explaining that algebra uses a language of numbers, variables, and symbols to represent quantitative relationships. The document then covers topics like equivalent expressions, simplifying expressions, evaluating expressions, and different types of polynomials. It aims to help students acquire skills in performing operations with polynomials and using those skills to simplify, evaluate, and solve algebraic expressions and problems.
Similar to Functions O level (20)
Quarter 1 - Illustrating and solving quadratic equations
Quarter 1 - Illustrating and solving quadratic equations
1.2 simplifying expressions and order of operations
1.2 simplifying expressions and order of operations
Recently uploaded
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingExcellence Foundation for South Sudan
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.DRUGS AND ITS classification slide share
Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...Nguyen Thanh Tu Collection
https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdclinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
How to Fix the Import Error in the Odoo 17
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Community pharmacy- Social and preventive pharmacy UNIT 5
Covered community pharmacy topic of the subject Social and preventive pharmacy for Diploma and Bachelor of pharmacy
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcSouth African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)Academy of Science of South Africa
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.Digital Artifact 1 - 10VCD Environments Unit
Digital Artifact 1 - 10VCD Environments Unit - NGV Pavilion Concept Design
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS Template 2023-2024 by: Irene S. Rueco
Recently uploaded (20)
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
Digital Artefact 1 - Tiny Home Environmental Design
Digital Artefact 1 - Tiny Home Environmental Design
Community pharmacy- Social and preventive pharmacy UNIT 5
Community pharmacy- Social and preventive pharmacy UNIT 5
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
Functions O level
- 2. How to evaluate an algebraic formula • RECAP • Example: given that 𝐹 = 𝑚(𝑣−𝑢) 𝑟2 , find the value of f where m = 80, v = 20 , u = 10 and r = 5. Solution: we substitute all given values of the letters accordingly. 𝐹 = 80(20−10) 52 𝐹 = 80(10) 25 𝐹 = 800 25 hence, we get F = 32 This concept is a base for further discussion
- 3. Functions of higher order We already have studied • Linear function 𝒇(𝒙) = 𝒎𝒙 + 𝒄 • Quadratic function 𝒇(𝒙) = 𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄 • There are more functions according to the degree of variable. Here we are going to evaluate different functions for the values given in the question. Example: Given that 𝒇 𝒙 = 𝟑𝒙 𝟐 − 𝟏, find 𝒇 𝟐 𝟏 𝟑 Solution: Substitute 𝐱 = 𝟐 𝟏 𝟑 in the function and solve it. 𝒇 𝟑 𝟏 𝟐 = 𝟑 𝟑 𝟏 𝟐 𝟐 − 𝟏 𝒇 𝟑 𝟏 𝟐 = 𝟑 𝟕 𝟑 𝟐 − 𝟏 𝒇 𝟑 𝟏 𝟐 = 𝟑 𝟒𝟗 𝟗 − 𝟏 𝒇 𝟑 𝟏 𝟐 = 𝟒𝟗 𝟑 − 𝟏 = 𝟒𝟔 𝟑 = 𝟏𝟓 𝟏 𝟑 (simplified answer) Activity: D-3 Exercise 2A Q1 & 2
- 4. PROBLEMS INVOLVING FUNCTIONS AND SIMULTANEOUS EQUATIONS Example: Given that the function h 𝒙 = 𝒄𝒙 + 𝒅 and h(2) = 6 and h(7) = 16, find the value of h and of d. Solution: 𝒉 𝟐 = 𝟐𝒄 + 𝒅 = 𝟔 (1) 𝒉 𝟕 = 𝟕𝒄 + 𝒅 = 𝟏𝟔 (2) By subtracting equation (1) from (2), we get 𝟓𝒄 = 𝟏𝟎 c = 𝟐 Substituting c = 𝟐 into equation 1 𝟐(𝟐) + 𝒅 = 𝟔 𝟒 + 𝒅 = 𝟔 Activity : Ex 2A Q 3, 4 & 5