Test yourself unit 2 foundation qs
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Unit 2 covers various formulas and mathematical concepts including: 1) Formulas for speed, opposites of square and cube, equations of straight lines, and rules of indices. 2) Parallel lines have the same gradient and a number in standard form has two parts with an exponent. 3) Procedures for adding, subtracting, multiplying, and dividing fractions as well as solving, simplifying, expanding, factorizing equations and changing subjects.
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Test yourself unit 2 foundation as
The document provides mathematical formulas, terms, and concepts. It includes definitions for speed, indices, parallel lines, standard form, fractions, inequalities, and sequences. It also lists steps for operations like adding and subtracting fractions, multiplying fractions, dividing fractions, expanding brackets, factorizing, and changing the subject of a formula.
Square roots
The document discusses square roots including:
1. Defining square roots and the radical symbol, and that the square root of a number can be positive or negative.
2. Explaining that the principal square root is always positive and how to identify squares of integers from 1 to 20.
3. Giving examples of simplifying square root expressions and identifying numbers as rational or irrational.
4. Discussing estimating square roots by finding what two consecutive integers a square root lies between.
5. Providing a word problem about finding the length of a wire supporting a tower using a square root formula.
Test yourself unit 3 foundation qs
This document provides formulas and definitions for geometry and math concepts covered in Unit 3. It includes formulas for calculating the area of shapes like triangles, parallelograms, trapezoids, and circles. It also includes formulas for volume of cubes, cuboids, prisms, cylinders, as well as definitions for terms like congruent, similar, perpendicular, and more. The document is a comprehensive study guide listing key formulas and explanations for foundational geometry and math topics.
2.4 Linear Functions
Identify whether a relation is a function
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Solving Quadratics
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the definitions and formulas for each form. It then explains how to graph each form by identifying key features of the equation, finding important points like the vertex, axis of symmetry, intercepts, and connecting points to sketch the parabolic curve. Graphing techniques include using the value of a to determine the opening direction, using b and c for standard form, using h and k for vertex form, and using p and q for intercepts form.
January 16, 2015
- The document outlines notes on slope and linear equations from a class. It includes definitions of slope, formulas for calculating slope between two points, and examples of finding the slope and equation of a line from graphical data or given information.
- Students will review slope formulas, work examples and practice problems, and complete class work 2.11 applying concepts of slope and linear equations. The notes provide guidance on finding slope, writing equations of lines in slope-intercept and standard form, and graphing lines on coordinate planes.
Rational Functions
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
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Test yourself unit 2 foundation as
The document provides mathematical formulas, terms, and concepts. It includes definitions for speed, indices, parallel lines, standard form, fractions, inequalities, and sequences. It also lists steps for operations like adding and subtracting fractions, multiplying fractions, dividing fractions, expanding brackets, factorizing, and changing the subject of a formula.
Square roots
The document discusses square roots including:
1. Defining square roots and the radical symbol, and that the square root of a number can be positive or negative.
2. Explaining that the principal square root is always positive and how to identify squares of integers from 1 to 20.
3. Giving examples of simplifying square root expressions and identifying numbers as rational or irrational.
4. Discussing estimating square roots by finding what two consecutive integers a square root lies between.
5. Providing a word problem about finding the length of a wire supporting a tower using a square root formula.
Test yourself unit 3 foundation qs
This document provides formulas and definitions for geometry and math concepts covered in Unit 3. It includes formulas for calculating the area of shapes like triangles, parallelograms, trapezoids, and circles. It also includes formulas for volume of cubes, cuboids, prisms, cylinders, as well as definitions for terms like congruent, similar, perpendicular, and more. The document is a comprehensive study guide listing key formulas and explanations for foundational geometry and math topics.
2.4 Linear Functions
Identify whether a relation is a function
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The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the definitions and formulas for each form. It then explains how to graph each form by identifying key features of the equation, finding important points like the vertex, axis of symmetry, intercepts, and connecting points to sketch the parabolic curve. Graphing techniques include using the value of a to determine the opening direction, using b and c for standard form, using h and k for vertex form, and using p and q for intercepts form.
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- The document outlines notes on slope and linear equations from a class. It includes definitions of slope, formulas for calculating slope between two points, and examples of finding the slope and equation of a line from graphical data or given information.
- Students will review slope formulas, work examples and practice problems, and complete class work 2.11 applying concepts of slope and linear equations. The notes provide guidance on finding slope, writing equations of lines in slope-intercept and standard form, and graphing lines on coordinate planes.
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This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
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Rational functions 13.1 13.2
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Determine vertical, horizontal, and oblique asymptotes of rational functions.
Graph rational functions
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This document provides steps for writing equations of lines in point-slope form, standard form, and finding perpendicular lines. It explains how to:
1) Find the slope between two integer points on a line using the slope formula.
2) Write the line equation in point-slope form using the slope and a point coordinate.
3) Convert the equation to standard form by distributing and adding/subtracting terms.
4) Find the y-intercept by setting x=0 and the slope using two points including the y-intercept.
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This document discusses how to graph rational functions by identifying key characteristics from the function expression. These include:
1) The y-intercept by setting x=0.
2) X-intercepts by setting the numerator equal to 0.
3) Vertical asymptotes by setting the denominator equal to 0.
4) Horizontal or slant asymptotes using rules based on the degrees of the numerator and denominator polynomials.
5) The graph by considering the intercepts, asymptotes, and a "sign property" that determines whether the graph is above or below the x-axis between intercepts/asymptotes. Examples are worked through step-by-step.
M17 t1 notes
The document provides step-by-step instructions for graphing rational functions. It explains that rational functions are fractions of polynomials, and defines key features like domains, zeros, vertical and horizontal asymptotes. Examples are given to demonstrate identifying these features and graphing the rational function based on them. Steps include finding intercepts, asymptotes, determining where the graph lies relative to the x-axis, and combining these elements to graph the full function.
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This document discusses rational functions and their graphs. It outlines 3 cases for rational functions based on the degree of the numerator and denominator. It also discusses identifying and plotting asymptotes, including vertical, horizontal, and slant asymptotes. Methods are provided for graphing rational functions based on finding asymptotes and plotting points around the asymptotes to determine the graph's behavior. Examples are worked through demonstrating these graphing techniques.
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- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
- The line is drawn solid for ≤ or ≥ and dashed for < or >. Shading the correct side of the line indicates the full solution set of the inequality.
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1. The document discusses absolute value functions and how to graph and write equations for them.
2. It explains that the graph of an absolute value function is V-shaped and defines the vertex and line of symmetry.
3. Examples are provided to demonstrate how to graph an absolute value function by finding the vertex and slope and plotting points, and how to write the equation of an absolute value function given its graph.
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Slope-intercept form is represented by the equation y=mx+b, where m is the slope and b is the y-intercept. There are three methods to find the slope: rise over run, change in y over change in x, and point-slope formula. The slope tells the rate of change, while the y-intercept is the initial value when x=0. To graph a line in slope-intercept form, you plot the y-intercept and then use the slope to trace successive points on the line by calculating the rise over run at each interval of x. Word problems can be modeled using this form by identifying the independent and dependent variables and finding their relationship through calculating slope from ordered pairs of known values.
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ICSE class X maths booklet with model paper 2015
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quadraticequations-111211090004-phpapp02.pptx
Quadratic equations are polynomial equations of the second degree that take the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using methods like factoring, completing the square, or the quadratic formula. The discriminant, b2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers.
Analytic Geometry Period 1
The document provides an overview of various topics in analytic geometry, including circle equations, distance equations, systems of two and three variable equations, linear inequalities, rational inequalities, and intersections of inequalities. It defines key concepts, provides examples of how to solve different types of problems, and notes things to remember when working with inequalities.
Short notes on mathematics
A parallelogram is a quadrilateral where opposite sides are equal and opposite angles are equal, and a diagonal of a parallelogram divides it into two congruent triangles.
presentation_quadraticequations-111211090004-phpapp02_1524500815_313961.pptx
Quadratic equations are polynomial equations of the second degree that can be written in the general form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: using the quadratic formula, factoring, or completing the square. The quadratic formula provides the exact solutions and can be used to solve any quadratic equation. Factoring and completing the square involve rewriting the equation in an equivalent form to reveal the solutions.
Maths
This document discusses linear equations in two variables. It defines linear equations in two variables as equations of the form ax + by = c, where a, b, and c are real numbers and a and b are not both zero. It explains that the graph of any linear equation in two variables is a straight line. It also categorizes different types of systems of linear equations based on the relationship between the lines: intersecting lines have a unique solution; coincident lines have an infinite number of solutions; and parallel lines have no solution. Methods for solving systems of linear equations like substitution, elimination, and graphing are also covered.
QUADRATIC.pptx
The document defines and provides examples of quadratic equations. It begins by stating that a quadratic equation is a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Roots of a quadratic equation are the values that make the equation equal to 0. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. The discriminant can be used to determine the nature of the roots.
Additional Mathematics Revision
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
Long and synthetic division
The document discusses different methods for dividing polynomials, including:
1) Dividing by a monomial by splitting the polynomial into fractions and reducing.
2) Performing long division of polynomials similar to long division of integers.
3) Using synthetic division as a shortcut for long division when the divisor is of the form x - k, where k is a number.
4) An example of using synthetic division to factor a polynomial completely when given one of its factors.
February 3, 2015
The document contains notes and examples about graphing linear inequalities on a coordinate plane. It discusses writing inequalities in y=mx+b form, determining the boundary line, and shading the appropriate region based on whether the inequality is <, ≤, >, or ≥. Key steps include solving the inequality for y, graphing the boundary line, and testing a point such as (0,0) to determine which side of the line to shade.
Algebra Tiles Pp Version 2
Algebra tiles can be used to model algebraic concepts in a concrete way to help students develop understanding. They represent numbers and variables that can be manipulated to demonstrate addition, subtraction, multiplication, division and other operations. Using the tiles allows students to see relationships and properties of algebra visually and physically that supports learning. The tiles have different colors and shapes to represent numbers, variables, exponents and operations.
linear equations in two variables
This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.
linear equation in 2 variables
1) A linear equation in two variables can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0.
2) The solution to a linear equation is the ordered pair that satisfies the equation when substituted into it.
3) Linear equations can be graphed on a Cartesian plane by plotting the solutions as points and connecting them.
Transformations
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
Similar to Test yourself unit 2 foundation qs (20)
linearequtionswithonevariable-160608051717 (1).pptx
linearequtionswithonevariable-160608051717 (1).pptx
presentation_quadraticequations-111211090004-phpapp02_1524500815_313961.pptx
presentation_quadraticequations-111211090004-phpapp02_1524500815_313961.pptx
Test yourself unit 2 foundation qs
- 2. 10 % divide by 10
- 3. Estimate means round and workout!
- 4. The x axis is called y=0 The y axis is called x=0
- 5. Speed = distance ÷ time
- 6. The opposite of square is square root The opposite of cube is cube root
- 7. y x y = y²
- 8. y + y = 2y
- 9. Equation of a straight line is y = +/- gradient x +/- Intercept
- 10. The Rules of indices a4 x a5 = a9 a7 ÷ a3 = a4 (a6)3 = a18
- 11. Parallel lines have the same gradient
- 12. A number in Standard Form is in two parts 3.6 x 106
- 13. To add and subtract fractions the denominators must be the same
- 14. To multiply fractions we multiply top by top bottom by bottom
- 15. Negative x negative = Positive Positive x negative = negative
- 16. To divide fractions we invert the second fraction and multiply
- 17. Solve means to find the value of x= or y=
- 18. Simplify means collect all the like terms.
- 19. Expand means remove the brackets
- 20. Factorise means put the brackets back in. Factor (outside bracket)
- 21. Change the subject means rearrange legally!!!
- 22. Expanding double brackets ( x + 4 )( x - 3 )
- 25. For sequences find the ‘goes up in’ and adjust.