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Functions
1. The document defines analytic plane quadrants and ordered pairs.
2. An ordered pair (a,b) contains two elements where a is the first component and b is the second. Equality of ordered pairs means the first elements are equal and the second elements are equal.
3. Notations for intervals are defined including closed, open, and half-open intervals using brackets [], parentheses (), and brackets with parentheses [).
Momentum
Momentum is defined as the product of an object's mass and velocity. It is represented by the symbol p. The impulse-momentum theorem states that the change in an object's momentum (Δp) is equal to the impulse (F×Δt) applied to it. Impulse is the average force applied over a time period.
Angles g7
An angle is formed by two lines that share a common point called the vertex. Angles are measured in degrees, with a full revolution being 360 degrees. A protractor can be used to both measure and construct angles. Angles are classified as acute (<90 degrees), right (90 degrees), obtuse (>90<180 degrees), or straight (180 degrees). Complementary angles sum to 90 degrees, while supplementary angles sum to 180 degrees.
Dilation
A dilation is a transformation that changes the size of a figure but not its shape. A scale factor describes how much a figure is enlarged or reduced, where for a dilation with scale factor k, the image of a point is found by multiplying each coordinate by k. If the scale factor k is greater than 1, it is an enlargement, and if k is less than 1, it is a reduction.
Conservation of momentum
This document discusses the law of conservation of momentum, which states that in an isolated system, the total momentum of all objects interacting will remain constant regardless of the forces between them. It provides examples of how conservation of momentum applies not just to collisions, but also to situations where momentum is transferred between objects without colliding, such as skaters pushing off each other or a person stepping out of a boat.
Fractions
A fraction represents a part of a whole or group. It consists of a numerator above a fraction bar and a denominator below. Examples include describing parts of a pizza that have been eaten or players on a team with yellow cards. Fractions can be proper if the numerator is smaller than the denominator, improper if the numerator is larger, or mixed numbers combining whole numbers and fractions. Equivalent fractions have the same value, while simplifying reduces fractions to their lowest terms.
Complex numbers
This document provides a lesson on complex numbers and using imaginary numbers to solve quadratic equations with complex roots. It begins with examples of simplifying expressions involving square roots of negative numbers by expressing them in terms of the imaginary unit i. Then it shows how to solve quadratic equations with imaginary solutions by taking square roots and expressing the solutions in terms of i. Examples are also provided of finding the complex zeros of quadratic functions by setting the functions equal to 0 and factoring. The document emphasizes that complex solutions come in conjugate pairs, and gives examples of finding the complex conjugate of solutions.
Circles g7
The document defines key terms related to circles such as diameter, radius, circumference, and area. It provides the formulas to calculate the circumference (C = πD) and area (A = πr^2) of a circle. Examples are given to find the circumference and area of various circles using these formulas.
Recommended
Functions
1. The document defines analytic plane quadrants and ordered pairs.
2. An ordered pair (a,b) contains two elements where a is the first component and b is the second. Equality of ordered pairs means the first elements are equal and the second elements are equal.
3. Notations for intervals are defined including closed, open, and half-open intervals using brackets [], parentheses (), and brackets with parentheses [).
Momentum
Momentum is defined as the product of an object's mass and velocity. It is represented by the symbol p. The impulse-momentum theorem states that the change in an object's momentum (Δp) is equal to the impulse (F×Δt) applied to it. Impulse is the average force applied over a time period.
Angles g7
An angle is formed by two lines that share a common point called the vertex. Angles are measured in degrees, with a full revolution being 360 degrees. A protractor can be used to both measure and construct angles. Angles are classified as acute (<90 degrees), right (90 degrees), obtuse (>90<180 degrees), or straight (180 degrees). Complementary angles sum to 90 degrees, while supplementary angles sum to 180 degrees.
Dilation
A dilation is a transformation that changes the size of a figure but not its shape. A scale factor describes how much a figure is enlarged or reduced, where for a dilation with scale factor k, the image of a point is found by multiplying each coordinate by k. If the scale factor k is greater than 1, it is an enlargement, and if k is less than 1, it is a reduction.
Conservation of momentum
This document discusses the law of conservation of momentum, which states that in an isolated system, the total momentum of all objects interacting will remain constant regardless of the forces between them. It provides examples of how conservation of momentum applies not just to collisions, but also to situations where momentum is transferred between objects without colliding, such as skaters pushing off each other or a person stepping out of a boat.
Fractions
A fraction represents a part of a whole or group. It consists of a numerator above a fraction bar and a denominator below. Examples include describing parts of a pizza that have been eaten or players on a team with yellow cards. Fractions can be proper if the numerator is smaller than the denominator, improper if the numerator is larger, or mixed numbers combining whole numbers and fractions. Equivalent fractions have the same value, while simplifying reduces fractions to their lowest terms.
Complex numbers
This document provides a lesson on complex numbers and using imaginary numbers to solve quadratic equations with complex roots. It begins with examples of simplifying expressions involving square roots of negative numbers by expressing them in terms of the imaginary unit i. Then it shows how to solve quadratic equations with imaginary solutions by taking square roots and expressing the solutions in terms of i. Examples are also provided of finding the complex zeros of quadratic functions by setting the functions equal to 0 and factoring. The document emphasizes that complex solutions come in conjugate pairs, and gives examples of finding the complex conjugate of solutions.
Circles g7
The document defines key terms related to circles such as diameter, radius, circumference, and area. It provides the formulas to calculate the circumference (C = πD) and area (A = πr^2) of a circle. Examples are given to find the circumference and area of various circles using these formulas.
Trigonometry by mstfdemirdag
Trigonometry involves measuring angles and relationships between sides and angles of triangles. There are six trigonometric ratios - sine, cosine, tangent, cotangent, secant and cosecant - that relate the measures of sides and angles. Angles can be measured in degrees or radians and converted between the two units. Important trigonometric identities relate the ratios to each other and allow trigonometric functions of combined angles to be simplified.
Functions by mstfdemirdag
This document discusses key concepts related to functions including:
- The domain of a function is the set of all real numbers for which the expression is defined as a real number.
- Two functions are equal if and only if their expressions and domains are equal.
- An even function satisfies f(-x) = f(x) and an odd function satisfies f(-x) = -f(x). Graphs of even functions are symmetric to the y-axis and odds are symmetric to the origin.
- Composition of functions f o g is defined as (f o g)(x) = f(g(x)). Inverse functions satisfy y = f(x) if and only if x =
Trigonometric functions
This document discusses trigonometric functions and their properties. It defines periodic functions and their periods. It describes the amplitude of sine and cosine functions as half the difference between the maximum and minimum values. It discusses how transformations of a, b, h, and k values can stretch, compress, reflect, translate and shift the graphs of sine and cosine functions and how these affect their amplitudes and periods. Examples are provided to demonstrate identifying amplitudes, periods, phase shifts from transformed trigonometric functions.
Quadratic functions
Quadratic functions are polynomial functions of the second degree involving a variable squared. They can be represented by the general form f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0. Quadratic functions are useful for modeling many real-world phenomena such as projectile motion, vibrations, and growth and decay.
Sequences and series
The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.
Pre geometry
This document provides an overview of basic geometry concepts including:
- Points, lines, planes, and space as the building blocks of geometry.
- Definitions of key terms like points, lines, rays, segments, planes, angles, and different types of angles.
- How to measure angles using a protractor and the units of degrees.
- Relationships between geometric figures like parallel and intersecting lines, coplanar points and lines, and different types of angles.
Solving linear equations in two
1) A first degree equation containing two variables is called a linear equation.
2) There are two methods for graphing linear equations: plotting points that satisfy the equation or using the intercept method of graphing.
3) A set of two or more equations is called a system of equations. There are two methods for solving systems of equations: the elimination method and the substitution method.
Natural numbers
The document defines and describes properties of operations on natural numbers. It defines the set of natural numbers N as the set containing 0, 1, 2, 3, etc. It describes properties of addition, subtraction, multiplication and division of natural numbers such as closure, commutativity, associativity, identity elements and distributivity. Key properties are that natural numbers are closed under addition and multiplication but not subtraction or division.
Logic
This document defines key concepts in logic and propositions:
1) A proposition is a statement that is either true or false, such as "Dogs have six legs" or "2+3=5".
2) The truth value of a proposition is either 1 (true) or 0 (false).
3) Two propositions are equivalent if they have the same truth value. The negation of a proposition, denoted by p', is its opposite truth value.
4) A compound proposition combines two or more simple propositions using logical connectives like "and", "or", or "if/then".
Density
Density is defined as mass per unit volume. The SI unit for density is kg/m3, but g/cm3 is more commonly used. To calculate density, mass is measured using a scale and volume is measured using a ruler or graduated cylinder for irregular objects. Density can be used to differentiate between substances and is a characteristic property of matter. The density of a mixture is calculated by dividing the total mass of the mixture by the total volume.
Mechanics
The document provides formulas and examples for calculating acceleration, velocity, displacement, and time given constant acceleration. It defines acceleration as the rate of change of velocity with time and provides the kinematic equations used to solve constant acceleration problems involving initial velocity, final velocity, acceleration, displacement, and time. Examples include calculating the time, velocity, or displacement of objects accelerating uniformly such as cars, planes, and treadmills.
Divisibility
This document discusses divisibility rules for numbers 2 through 11. It provides examples and explanations of divisibility rules for each number, such as a number is divisible by 2 if its last digit is even, and it is divisible by 3 if the sum of its digits is divisible by 3. The document uses examples such as determining which numbers are divisible by 4 or finding possible digit values for a number to be divisible by 9.
Free fall
All objects accelerate at the same rate when falling near the Earth's surface due to gravity. This constant downward acceleration is denoted as "g" and is about 9.81 m/s^2 on Earth. Free fall acceleration causes objects to fall with increasing downward velocity over time regardless of whether the motion is upward or downward. The document then provides example problems calculating velocity, time, and displacement during free fall.
Prime numbers and factorization
This document defines key concepts related to factors and prime numbers of natural numbers. It explains that factors are numbers that divide a number with no remainder, and that prime numbers only have two factors - 1 and itself. Composite numbers have more than two factors. The prime factors of a number are its factors that are prime numbers. Writing a number as the product of its prime factors is called prime factorization.
Exponents
The document discusses exponents and exponential expressions. It defines exponential expressions as involving a base number being multiplied by itself a certain number of times, called the exponent. It provides examples of exponential expressions and how they are read out loud. It also outlines some properties of exponential expressions, such as all powers of positive numbers being positive and the behavior of even and odd powers of negative numbers. Finally, it discusses like terms in exponential expressions and an identity property involving exponents.
Motion in two dimensions
The document discusses projectile motion, which refers to two-dimensional motion of objects subject to gravity. Examples of projectiles include balls and arrows. Projectiles follow a parabolic trajectory path. The range of a projectile is the maximum horizontal distance it can cover while returning to its original launch height. To analyze projectile motion, the initial velocity vector is resolved into horizontal and vertical components, and kinematic equations are applied to describe motion in each dimension throughout flight.
Force
The document defines force and describes its key properties, including that it can start or stop motion, change speed or direction of motion, or change the shape or size of an object. It also defines different types of forces like gravitational, magnetic, electrostatic, tension, and frictional forces. The document explains that force is measured in Newtons and discusses action-reaction forces.
Radicals
This document discusses working with radicals and square roots. It provides examples of properties of square roots, such as if a2 = b then a is the square root of b where a and b are greater than or equal to 0. The document uses multiple examples to illustrate properties and concepts related to working with radicals and square roots.
Relative motion
Relative motion depends on frame of reference. Observers in different frames may measure different velocities for the same object. Several examples are given of calculating relative velocities between objects in motion, including: a boat crossing a river at different velocities than the moving water; a baseball thrown on a moving train; a spy running on a moving aircraft carrier; a ferry crossing a moving river; and a dog moving inside a moving supply truck.
Radicals part 2
The document discusses radicals, rationalizing denominators, and binomial expressions. It defines rationalizing a denominator as changing the denominator from an irrational number to a rational number without changing the value of the fraction. It also defines conjugates as two binomial expressions whose first terms are equal and last terms are opposite, such as a + b and a - b. Examples are provided of rationalizing denominators and working with conjugates.
More Related Content
More from mstf mstf
Trigonometry by mstfdemirdag
Trigonometry involves measuring angles and relationships between sides and angles of triangles. There are six trigonometric ratios - sine, cosine, tangent, cotangent, secant and cosecant - that relate the measures of sides and angles. Angles can be measured in degrees or radians and converted between the two units. Important trigonometric identities relate the ratios to each other and allow trigonometric functions of combined angles to be simplified.
Functions by mstfdemirdag
This document discusses key concepts related to functions including:
- The domain of a function is the set of all real numbers for which the expression is defined as a real number.
- Two functions are equal if and only if their expressions and domains are equal.
- An even function satisfies f(-x) = f(x) and an odd function satisfies f(-x) = -f(x). Graphs of even functions are symmetric to the y-axis and odds are symmetric to the origin.
- Composition of functions f o g is defined as (f o g)(x) = f(g(x)). Inverse functions satisfy y = f(x) if and only if x =
Trigonometric functions
This document discusses trigonometric functions and their properties. It defines periodic functions and their periods. It describes the amplitude of sine and cosine functions as half the difference between the maximum and minimum values. It discusses how transformations of a, b, h, and k values can stretch, compress, reflect, translate and shift the graphs of sine and cosine functions and how these affect their amplitudes and periods. Examples are provided to demonstrate identifying amplitudes, periods, phase shifts from transformed trigonometric functions.
Quadratic functions
Quadratic functions are polynomial functions of the second degree involving a variable squared. They can be represented by the general form f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0. Quadratic functions are useful for modeling many real-world phenomena such as projectile motion, vibrations, and growth and decay.
Sequences and series
The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.
Pre geometry
This document provides an overview of basic geometry concepts including:
- Points, lines, planes, and space as the building blocks of geometry.
- Definitions of key terms like points, lines, rays, segments, planes, angles, and different types of angles.
- How to measure angles using a protractor and the units of degrees.
- Relationships between geometric figures like parallel and intersecting lines, coplanar points and lines, and different types of angles.
Solving linear equations in two
1) A first degree equation containing two variables is called a linear equation.
2) There are two methods for graphing linear equations: plotting points that satisfy the equation or using the intercept method of graphing.
3) A set of two or more equations is called a system of equations. There are two methods for solving systems of equations: the elimination method and the substitution method.
Natural numbers
The document defines and describes properties of operations on natural numbers. It defines the set of natural numbers N as the set containing 0, 1, 2, 3, etc. It describes properties of addition, subtraction, multiplication and division of natural numbers such as closure, commutativity, associativity, identity elements and distributivity. Key properties are that natural numbers are closed under addition and multiplication but not subtraction or division.
Logic
This document defines key concepts in logic and propositions:
1) A proposition is a statement that is either true or false, such as "Dogs have six legs" or "2+3=5".
2) The truth value of a proposition is either 1 (true) or 0 (false).
3) Two propositions are equivalent if they have the same truth value. The negation of a proposition, denoted by p', is its opposite truth value.
4) A compound proposition combines two or more simple propositions using logical connectives like "and", "or", or "if/then".
Density
Density is defined as mass per unit volume. The SI unit for density is kg/m3, but g/cm3 is more commonly used. To calculate density, mass is measured using a scale and volume is measured using a ruler or graduated cylinder for irregular objects. Density can be used to differentiate between substances and is a characteristic property of matter. The density of a mixture is calculated by dividing the total mass of the mixture by the total volume.
Mechanics
The document provides formulas and examples for calculating acceleration, velocity, displacement, and time given constant acceleration. It defines acceleration as the rate of change of velocity with time and provides the kinematic equations used to solve constant acceleration problems involving initial velocity, final velocity, acceleration, displacement, and time. Examples include calculating the time, velocity, or displacement of objects accelerating uniformly such as cars, planes, and treadmills.
Divisibility
This document discusses divisibility rules for numbers 2 through 11. It provides examples and explanations of divisibility rules for each number, such as a number is divisible by 2 if its last digit is even, and it is divisible by 3 if the sum of its digits is divisible by 3. The document uses examples such as determining which numbers are divisible by 4 or finding possible digit values for a number to be divisible by 9.
Free fall
All objects accelerate at the same rate when falling near the Earth's surface due to gravity. This constant downward acceleration is denoted as "g" and is about 9.81 m/s^2 on Earth. Free fall acceleration causes objects to fall with increasing downward velocity over time regardless of whether the motion is upward or downward. The document then provides example problems calculating velocity, time, and displacement during free fall.
Prime numbers and factorization
This document defines key concepts related to factors and prime numbers of natural numbers. It explains that factors are numbers that divide a number with no remainder, and that prime numbers only have two factors - 1 and itself. Composite numbers have more than two factors. The prime factors of a number are its factors that are prime numbers. Writing a number as the product of its prime factors is called prime factorization.
Exponents
The document discusses exponents and exponential expressions. It defines exponential expressions as involving a base number being multiplied by itself a certain number of times, called the exponent. It provides examples of exponential expressions and how they are read out loud. It also outlines some properties of exponential expressions, such as all powers of positive numbers being positive and the behavior of even and odd powers of negative numbers. Finally, it discusses like terms in exponential expressions and an identity property involving exponents.
Motion in two dimensions
The document discusses projectile motion, which refers to two-dimensional motion of objects subject to gravity. Examples of projectiles include balls and arrows. Projectiles follow a parabolic trajectory path. The range of a projectile is the maximum horizontal distance it can cover while returning to its original launch height. To analyze projectile motion, the initial velocity vector is resolved into horizontal and vertical components, and kinematic equations are applied to describe motion in each dimension throughout flight.
Force
The document defines force and describes its key properties, including that it can start or stop motion, change speed or direction of motion, or change the shape or size of an object. It also defines different types of forces like gravitational, magnetic, electrostatic, tension, and frictional forces. The document explains that force is measured in Newtons and discusses action-reaction forces.
Radicals
This document discusses working with radicals and square roots. It provides examples of properties of square roots, such as if a2 = b then a is the square root of b where a and b are greater than or equal to 0. The document uses multiple examples to illustrate properties and concepts related to working with radicals and square roots.
Relative motion
Relative motion depends on frame of reference. Observers in different frames may measure different velocities for the same object. Several examples are given of calculating relative velocities between objects in motion, including: a boat crossing a river at different velocities than the moving water; a baseball thrown on a moving train; a spy running on a moving aircraft carrier; a ferry crossing a moving river; and a dog moving inside a moving supply truck.
Radicals part 2
The document discusses radicals, rationalizing denominators, and binomial expressions. It defines rationalizing a denominator as changing the denominator from an irrational number to a rational number without changing the value of the fraction. It also defines conjugates as two binomial expressions whose first terms are equal and last terms are opposite, such as a + b and a - b. Examples are provided of rationalizing denominators and working with conjugates.