Algebra 1 Revision: brackets, indices
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The document outlines key concepts and skills to be learned in an algebra lesson, including: - Simplifying algebraic expressions by collecting like terms - Expanding brackets - Multiplying algebraic expressions It then provides examples of simplifying algebraic expressions, expanding brackets, finding perimeters and areas of shapes, evaluating expressions, and converting between Fahrenheit and Celsius temperatures.
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3. Each unit lists the corresponding textbook chapters and New York State Common Core Learning Standards addressed. Common assessments are also included.
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The document provides an overview of quadratic functions including definitions of key terms like quadratic function, parabola, quadratic equation, and vertex form. It discusses how to graph quadratic functions, solve quadratic equations using methods like the quadratic formula, and find the maximums and minimums of quadratic functions. It includes examples of using these techniques to solve word problems involving quadratic applications. The document aims to teach students about the basic concepts of quadratic functions.
Sample fin
1) Simplify expressions and evaluate given values for variables. Solve equations algebraically and graphically.
2) Multiply, expand, simplify, and factor polynomial expressions. Determine if expressions are factorable.
3) Solve systems of linear equations and word problems involving area calculations to find unknown values.
Algebra unit 8.7
This document provides examples and explanations for factoring special cases of polynomials, including perfect square trinomials and the difference of two squares. It includes examples of recognizing, factoring, and explaining if expressions are in one of these forms. One example problem finds the perimeter of a garden given its area as a factored expression and evaluates the perimeter for a given value of x.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
Prac 3 lagrange's thm
This document provides instructions for verifying Lagrange's theorem using Geogebra. Students are asked to:
1. Open the "Verification of Rolle's theorem" applet and save it with their name.
2. Observe the function f(x)=x^3 - 3x^2 + 4 is defined on [-1,3] and ensure it is continuous and differentiable.
3. Record observations of the point A with various slopes between the intervals [-1,1] and [1,3] and determine if the slope equals the slope of the chord BC.
4. The points x1 and x2 that satisfy Lagrange's theorem are where the tangent slope equals
Prac 3 lagrange's thm
This document provides instructions for verifying Lagrange's theorem using Geogebra. Students are asked to:
1) Open the "Verification of Rolle's theorem" applet and save it with their name.
2) Observe the function f(x)=x^3-3x^2+4 defined on [-1,3] and ensure continuity and differentiability.
3) Record observations of the point A with various slopes and compare the tangent slope to the slope of the chord BC at different intervals.
4) The points that satisfy Lagrange's theorem are where the tangent slope equals the chord BC slope between the endpoints.
Distributive property
The students will learn to use the distributive property to simplify expressions by distributing terms being multiplied to terms inside parentheses. The distributive property distributes the number outside the parentheses to each term inside. Examples are provided to demonstrate distributing terms and combining like terms to simplify expressions.
Section 1.5 distributive property (algebra)
The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction, such as 5(x + 7) = 5x + 35. It also discusses like terms and how they can be combined when simplifying expressions.
Distributive
The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction. It defines terms, coefficients, and like terms. It then gives examples of simplifying expressions by combining like terms.
Quadratic equations
The document discusses various methods for solving quadratic equations, including factoring, square root method, completing the square, and the quadratic formula. It also defines terms related to quadratic equations like discriminant, complex numbers, and pure and complete quadratic equations. The objectives are for students to learn to solve different types of quadratic equations and problems using these various methods and concepts.
Topic 14 algebra
The document discusses rules for indices and factorizing algebraic expressions. It provides examples of:
- Multiplying and dividing terms with the same base using index rules like ab × ac = ab+c and ab ÷ ac = ab-c.
- Expanding single and double brackets by distributing terms.
- Finding common factors to group like terms.
- Factorizing quadratics and using differences of squares.
- Solving equations set equal to zero by factorizing.
Hprec2 5
This document provides an overview of solving inequalities, including:
1) The principles for solving inequalities are similar to equations but the direction of the inequality changes when multiplying or dividing by a negative number.
2) Interval notation is used to represent the solutions of inequalities on the number line.
3) To solve inequalities, determine the zeros of the function and where the graph is above or below the x-axis to identify the intervals of solutions.
Quadratic Function Presentation
The document defines and explains key concepts regarding quadratic functions including:
- The three common forms of quadratic functions: general, vertex, and factored form
- How to find the x-intercepts, y-intercept, and vertex of a quadratic function
- Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula
- How to graph quadratic functions by identifying intercepts and the vertex
Module 11 topic 1 notes(2)
The document discusses graphing and solving quadratic equations. It defines key terms like parabolas, vertices, and axes of symmetry. It provides examples of using the quadratic formula to find the vertex and shows how to factor and set factors equal to zero to solve quadratic equations and find their roots.
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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.
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.
5) logarithms graphs
This document provides an overview of exponential and logarithmic functions:
1) It explores properties of exponential functions with the forms y = ax, where a > 1 represents exponential growth and 0 < a < 1 represents exponential decay. Key aspects like asymptotes and y-intercepts of these graphs are discussed.
2) Tables are completed and graphs are drawn to represent exponential growth and decay functions.
3) The natural logarithm function ln(x) is introduced and its graph is sketched using a calculator. Common features that logarithmic graphs share with exponential graphs are described.
4) Examples of simplifying logarithmic expressions and solving logarithmic equations are presented.
4) logarithms laws
This document discusses logarithmic laws and their proofs. It contains:
- Exercises to verify logarithmic laws like loga(xy) = loga(x) + loga(y) and nloga(x) = loga(x)n
- Proofs of these laws using the definition of logarithms and index form
- More exercises for students to solve involving simplifying logarithmic expressions using these laws
The goal is for students to understand and be able to apply the key logarithmic laws through worked examples and practice questions.
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.
2) exponential growth and decay
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
1) exponential equations
This document discusses exponential functions and equations. It explores the properties of exponential functions by examining their graphs and understanding exponential growth and decay. It also covers how to solve problems leading to exponential equations by learning about bases, exponents, powers, and index rules. The document teaches how to evaluate, write in different bases, identify exponential equations, and solve exponential equations. It includes exercises to solve from the attached document on exponential equations and functions.
6) simultaneous and problems
This document discusses finding the intersection points between a line and parabola by solving simultaneous equations, finding the value of k that makes a line tangent to a parabola, and finding the maximum area of a rectangle with a perimeter of 28 cm.
5) quadratic equations
This document discusses solving quadratic equations by different methods such as factorisation, using the quadratic formula, and completing the square. It explains how to determine the number of solutions a quadratic equation has based on the sign of its discriminant, namely:
- If the discriminant is positive, there are two real roots
- If the discriminant is zero, there is one repeated root
- If the discriminant is negative, there are no real roots.
Examples are provided to illustrate calculating the discriminant and determining the number and type of roots.
4) quadratic functions factorised form
This document discusses quadratic functions and their properties. It explains how to express quadratics in different forms like factored, general, and vertex form. It provides examples of finding the zeros, line of symmetry, and vertex of various quadratic functions. Students are asked to work through examples of sketching graphs, factorizing expressions, solving equations, and rewriting functions in different forms to investigate the properties of quadratic functions and their relationships between algebraic and graphical representations.
3) quadratic vertex form
The document discusses quadratic functions and their graphs. It aims to explore the properties of quadratic functions and express them in different forms, including vertex form. It provides examples of writing quadratic functions in vertex form given the vertex, as well as writing the function in general form given its vertex. The key points are that the vertex form of a quadratic function y = a(x - h)2 + k shows the vertex is (h, k) and the line of symmetry is x = h. Converting between the vertex and general forms allows finding the maximum point of a quadratic function.
2) quadratics gral form
This document discusses properties of quadratic functions and their graphs. It explores expressing quadratic functions in different forms, sketching graphs of basic quadratic functions, and investigating the effects of changing coefficients on the graph. The key aspects covered are the relationship between the coefficients and the graph properties like the vertex, line of symmetry, and y-intercept. Examples of matching functions to graphs and writing functions in standard form are provided.
1) quadratic functions
The document discusses quadratic functions and their graphs. It explores expressing quadratic functions in different forms, and transforming quadratic functions and their graphs through translations, stretches, and other changes. Key aspects covered include the vertex and line of symmetry of parabolas, and how changing the coefficients affects the graph shape and position. Examples of specific quadratic functions are analyzed and graphed.
Y9 algebra 1 Simultaneous Equations byElimination Method
This document discusses two methods for solving simultaneous equations: the elimination method and the substitution method. It provides examples of using the elimination method to solve several systems of simultaneous equations, showing the steps to eliminate one variable and solve for the other variables. The document concludes by providing two word problems as examples and assigning practice problems from the book.
Y9 algebra 1 simultaneous equations by substitution method
The document provides examples of solving simultaneous equations by substitution. It includes examples of setting up two equations representing relationships between variables, substituting one equation into the other to isolate a single variable, and solving for the value of that variable. It then uses the solved variable to find the value of the other variable.
Y9 algebra 1 Change of Subject
By the end of the lesson, students should be able to change the subject of a formula. The document provides examples of changing the subject of formulas for perimeter of a circle, cost of a taxi, volume of a cone, and other formulas. Students are asked to make different variables the subject of provided formulas, such as making d, a, m, h, or x the subject.
Y9 algebra 1 Inequalities
This document provides an overview of inequalities and how to solve them algebraically. It discusses representing inequalities on number lines, and the rules for adding, subtracting, multiplying, and dividing terms when solving inequalities, such as changing the sign of the inequality when multiplying or dividing by a negative number. Examples are provided of solving inequalities algebraically and representing the solutions on number lines.
Y9 algebra 1 Collecting Terms & Expanding Brackets
The document outlines key concepts and skills to be learned in an algebra lesson, including:
- Simplifying algebraic expressions by collecting like terms
- Expanding brackets
- Multiplying algebraic expressions
It then provides examples of simplifying algebraic expressions, expanding brackets, finding perimeters and areas of shapes, evaluating expressions, and converting between temperatures in Fahrenheit and Celsius.
Y9 algebra 1 Linear Equations
This document provides an overview of solving linear equations, including equations involving brackets, fractions, and changing the subject of a formula. It presents several example equations and step-by-step solutions showing how to isolate the variable. Students are reminded to check their answers by substituting them back into the original equation. The document also introduces solving equations with fractions using the lowest common denominator approach.
Ib maths sl chain rule
The document discusses applying the chain rule to find derivatives of composite functions. It provides examples of identifying composite functions and using the chain rule formula to take derivatives. It lists practice problems for students to find derivatives of various composite functions using the chain rule, including functions with exponents like (x3 + 1)8, as well as trigonometric functions like y = sin(4x).
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