Module 4 topic 1 part 3 notes
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This document defines key terms related to relations and functions, including: - A relation is a set of ordered pairs. The domain is the set of x-values and the range is the set of y-values. - A relation is a function if each x-value is mapped to only one y-value. This can be determined using the vertical line test or by checking if x-values are repeated. - Graphs where a vertical line crosses the relation more than once are not functions, while graphs where it only crosses once are functions.
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Relations are sets of ordered pairs that represent connections between elements. A relation is a function if each element in the domain (input) is uniquely mapped to an element in the range (output). This can be tested using the vertical line test - if a vertical line can pass through the graph of a relation more than once, it is not a function. The domain of a relation is the set of all first elements of each ordered pair, while the range is the set of all second elements. A relation is a function if its domain does not contain repeating elements.
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The document discusses functions and relations. It defines a function as a set of ordered pairs where each x-value is paired with only one y-value. A relation is a function only if each x-value is paired with exactly one y-value; if an x-value is paired with more than one y-value, it is not a function. It then asks the reader to identify which of several relations presented are functions based on this definition. It also gives some examples of evaluating simple functions.
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The document discusses relations and functions. It defines key terms like relation, domain, range, input, output, and function. A relation is a set of ordered pairs, while a function is a special type of relation where each input is paired with exactly one output. The domain is the set of all inputs and the range is the set of all outputs. Several examples are provided to demonstrate how to identify the domain and range of relations, represent relations using ordered pairs and graphs, and determine if a relation qualifies as a function using the vertical line test.
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This document discusses functions and how to determine if a relation is a function. It defines a function as a relation where each element of the domain maps to exactly one element of the range. It provides examples of tables, coordinate points, and graphs, and explains how to use the vertical line test to determine if a graph represents a function. It also discusses how to determine if a relation is a function algebraically by solving for the output variable and checking if there is only one output for each input.
PPt on Functions
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
Functions
A function is a relation where each element of the domain is paired with exactly one element of the range. The key characteristics are that there is one and only one output for each input. Some examples are provided to demonstrate determining if a relation represents a function based on this definition. The vertical line test is also introduced as a visual way to determine if a graph depicts a function. Finally, examples are given of evaluating functions for given inputs using function notation.
relationsandfunctionsupdated-140102120840-phpapp01.ppt
This document discusses relations and functions. It provides examples of how to classify relations as functions or not based on whether they satisfy the vertical line test and one-to-one correspondence. Students are asked to complete activities identifying domains and ranges of relations, graphing relations to apply the vertical line test, and determining if equations represent functions. The assignment is to copy equations on a sheet of paper and identify if they represent functions.
Chapter 3 Solving By Graph
This document provides information about functions and relations. It defines key terms like domain, range, and function. A relation is a mapping of inputs to outputs, while a function only maps each input to one output. Relations and functions can be represented using ordered pairs and graphed on a coordinate plane. The vertical line test can determine if a relation is a function by checking if any vertical line passes through more than one point. Functions can be represented by equations and evaluated using function notation like f(x).
Intro to Functions_Domain and Range.ppt
This document defines key concepts related to functions, including:
- A relation pairs inputs and outputs, while a function ensures each input has exactly one output.
- The domain is the set of all input values, and the range is the set of all output values.
- Functions can be represented graphically or through equations, with the domain and range specifying the sets of valid inputs and outputs.
Group H Presentation.pptx
This document discusses functions and their graphs. It defines relations, functions, and how to write and graph relations and functions. It discusses domain and range, and how to identify whether a relation is a function using the vertical line test. It provides examples of how to graph linear and non-linear equations, evaluate functions, and find the domain and range of functions.
Functions.ppt
1. A function is a relation where each input is mapped to exactly one output. It is not a function if one input has more than one output.
2. For a relation to be a function, every input must have an output and two different inputs cannot have the same output. One input can only have one output.
3. The vertical line test can be used to visually check if a relation is a function - if any vertical line intersects the graph at more than one point, it is not a function.
power point presentation on genmath_lesson1_2_.pptx
The document defines key terms related to functions including domain, range, and the relationship between independent and dependent variables. It provides examples of functions represented as sets of ordered pairs, equations, graphs, and tables. It discusses the vertical line test for determining if a relation represents a function. It also explains function notation and how to evaluate functions by substituting values for the independent variable.
1FUNCTIONS.pptx
1. The document discusses functions and how they can be represented through ordered pairs, tables of values, graphs, and equations.
2. A function is a relation where each element in the domain corresponds to exactly one element in the range.
3. Examples show how to determine if a relation represents a function using the vertical line test on graphs or by analyzing sets of ordered pairs.
Similar to Module 4 topic 1 part 3 notes (20)
relationsandfunctionsupdated-140102120840-phpapp01.ppt
relationsandfunctionsupdated-140102120840-phpapp01.ppt
relationsandfunctionslessonproper-160929053921.pdf
relationsandfunctionslessonproper-160929053921.pdf
power point presentation on genmath_lesson1_2_.pptx
power point presentation on genmath_lesson1_2_.pptx
More from Annie cox
Heating of the earth - 2
1. There are three main methods of heat transfer: conduction, convection, and radiation.
2. The Earth is heated through radiation from the sun, with some energy being absorbed, some reflected, and some transmitted through substances like water and air.
3. Factors like land/water distribution, altitude, latitude, and cloud cover contribute to geographic and climatic differences in surface temperatures around the world.
Air pressure
Air pressure is exerted equally in all directions by the air around us. It can be measured using a barometer, with units of mmHg or atm. Rapid changes in air pressure often indicate approaching weather systems - falling pressure means storms while rising pressure means clearing weather. Winds are created by differences in air pressure between areas. Global wind patterns are caused by uneven heating of the Earth, with the hottest air at the equator rising and causing circulation of air masses. High and low pressure systems form based on temperature and moisture levels, influencing wind direction.
Heating of the earth
1. There are three main methods of heat transfer: conduction, convection, and radiation.
2. The Earth is heated through radiation from the sun, with some energy being absorbed, some reflected, and some transmitted through substances like water and air.
3. Temperatures vary based on factors like proximity to water vs. land, altitude, latitude, and cloud cover, with land experiencing greater temperature ranges than water and higher latitudes being cooler on average.
Air pressure
Air pressure is exerted equally in all directions by the air around us. It can be measured using a barometer, with units of mmHg or atm. Rapid changes in air pressure often indicate approaching weather systems - falling pressure means storms while rising pressure means clearing weather. Winds are created by differences in air pressure between areas. Global wind patterns are caused by uneven heating of the Earth, with the hottest air at the equator rising and causing circulation of air masses. High and low pressure systems form based on temperature and moisture levels, influencing wind direction.
Unit 2 post assessment
The document provides 20 math word problems that require solving equations for a variable. It works through each problem step-by-step showing the work and reasoning to isolate the variable. Common steps include distributing terms, combining like terms, adding/subtracting the same quantity to both sides of the equation, and dividing/multiplying both sides by the same non-zero quantity. The final problems involve multiple steps to solve more complex equations.
Atmosphere lesson
The atmosphere is divided into four main layers - the troposphere, stratosphere, mesosphere, and thermosphere. Each layer is defined by changes in temperature. The troposphere extends from Earth's surface to around 12 km and contains around 90% of the atmosphere. The stratosphere extends from 12-50 km and contains the ozone layer which absorbs harmful UV radiation. Air is composed primarily of nitrogen (78%) and oxygen (21%).
Chapter 6 maps
Maps have evolved over time to become more accurate representations of the world. Early maps from 4000 BC showed where people lived and tax boundaries, while Ptolemy created the first world map in 150 AD using Greek and Roman cartography. In the Middle Ages, maps reflected cultural beliefs and guided pilgrims. Modern technology like GPS and GIS has led to highly accurate digital maps through satellite imagery and data analysis.
Chapter 2 current issues
This document provides an overview of current environmental issues facing the Earth. It discusses how human population growth and industrialization have negatively impacted the environment through increased consumption of non-renewable resources, pollution, and loss of biodiversity. As human societies developed from hunter-gatherers to agricultural to industrial, their demands on the environment expanded and dependence on fossil fuels rose, resulting in problems like pollution and depletion of resources. The core issues discussed are overconsumption, pollution, and biodiversity loss due to rising human populations. The document also introduces the concept of sustainability and the goal of creating a sustainable society that does not exceed the Earth's limits.
Chapter 14 – water resources
The document discusses water resources and water treatment. It explains that most of Earth's water is ocean water, while freshwater comes from surface water, groundwater, and the water cycle. It then describes how surface water forms rivers and lakes, groundwater forms aquifers, and how water is used residentially, industrially, and agriculturally. Issues like overdraft and habitat degradation are discussed, along with water treatment processes like desalination, purification through screening, sedimentation, filtration, aeration, and sterilization.
Chapter 22 – astronomical technology
The document summarizes different types of telescopes and space probes used to explore the universe, and some of their key contributions. It discusses radio, infrared, visible light, ultraviolet, X-ray, and gamma ray telescopes, as well as notable space telescopes like Hubble and the upcoming James Webb Space Telescope. It also outlines various space probes sent to study planets, moons, comets and more, providing images to advance scientists' understanding of the solar system. Finally, it lists some technologies that were developed for space exploration but later found applications on Earth.
Chapter 21 stars
1. Stars form from dense clouds of gas and dust called nebulae when gravity overcomes pressure, causing the cloud to collapse and heat up. Nuclear fusion then begins, forming a star.
2. Stars exist in different stages depending on their mass. Medium stars become red giants then white dwarfs. Massive stars explode as supernovae, leaving neutron stars or black holes.
3. Astronomers use tools like spectroscopy, star charts, and parallax to determine properties of stars like composition, temperature, brightness, and distance. This helps classify stars and map their life cycles.
Chapter 20 planetary motion
The document discusses planetary motion and the movement of celestial bodies. It describes how early astronomers like Ptolemy, Copernicus, and Kepler developed models of the solar system. Johannes Kepler developed his three laws of planetary motion, establishing that planets orbit the sun in ellipses. Isaac Newton later explained that gravity is the force that causes planets to orbit the sun. The document also discusses the Earth's rotation, revolution, and axial tilt, which cause seasons. It describes the moon's orbit and how its phases, tides, and eclipses occur due to the positions of the Earth, moon, and sun.
Chapter 19 – formation of the universe
The document summarizes theories about the formation and evolution of the universe:
1) The Big Bang theory proposes that approximately 13.8 billion years ago, all matter in the observable universe was condensed into an incredibly dense singularity that suddenly exploded, causing the universe to expand rapidly and form light elements.
2) As the universe expanded, gravity caused matter to condense into the first stars and galaxies. Observations of redshifts in light from distant galaxies provide evidence the universe continues expanding today.
3) Current theories suggest three possible ultimate fates for the universe depending on its overall density - continued expansion, eventual contraction, or a stable flat state.
Chapter 18 – air pollution and global changes
This document discusses various types of air pollution and their local and global effects. It describes how local pollution from activities like burning fossil fuels can cause issues worldwide through mechanisms like acid rain, ozone depletion, and global warming. Five major types of outdoor air pollutants are discussed - particulates, oxides, photochemical smog, hydrocarbons, and CFCs. The document also notes how air pollution affects human health, other organisms, and causes global problems through acid deposition, ozone layer thinning, and climate change due to increased greenhouse gases.
Chapter 17 – meteorology
Meteorologists use various data sources to predict the weather, but it is difficult to always be correct due to rapidly changing atmospheric conditions. Data comes from surface weather stations, weather balloons, satellites, radars, and computer models. Air masses and fronts influence weather, and storms form when different air masses meet. Hurricanes require specific conditions like warm ocean water. While meteorology studies the atmosphere, tiny differences in data can lead to different model results, so multiple models are used to predict daily weather.
Chapter 16 – the atmosphere and weather
1. The document discusses the four layers of Earth's atmosphere and how factors like temperature, pressure, wind, and moisture interact to cause weather.
2. It explains that the lowest layer, the troposphere, contains almost all of the atmosphere's mass and water and is where weather occurs.
3. The stratosphere contains the protective ozone layer. The thermosphere is where satellites orbit and temperatures can reach 2000 degrees Celsius.
4. Weather results from interactions of heat, air pressure, wind, and moisture. Unequal heating causes pressure differences that create wind, and wind circulations around the globe.
Chapter 15 – water quality and pollution
The document discusses various types of water pollution including chemical, thermal, radioactive, sediments, and sewage. The largest sources of water pollution are agricultural runoff and sewage. Developing countries often lack proper sewage treatment facilities. Water pollution has been an issue for centuries but legislation and enforcement have been weak, allowing the problem to continue. The top source of water pollution in North Carolina is stormwater runoff carrying various pollutants. Solutions include limiting development near waterways and preserving wetlands to naturally filter water.
Chapter 14 – water resources
The document discusses water resources and water treatment. It explains that most of Earth's water is ocean water, while freshwater comes from surface water, groundwater, and the water cycle. It then describes how surface water flows in rivers and collects in lakes and reservoirs, and how groundwater flows through permeable rock layers and can be accessed through wells. The document also outlines water use for residential, industrial, and agricultural purposes, as well as problems like overdraft and habitat degradation. It concludes by explaining processes for desalination and water purification to remove salts and contaminants, making water safe for drinking.
Chapter 13 – oceans and beaches
The document summarizes how oceans shape our lives by discussing the distribution of water on Earth, properties of ocean water, ocean motions, shoreline formations, and the barrier islands of North Carolina's Outer Banks. It explains that barrier islands formed due to rising sea levels, a large sand supply, a gently sloping coast, and wave energy. Humans are increasing the rate of barrier island migration through armoring and development, which reduces available sand and increases erosion risks. Ultimately, one cannot have buildings and beaches in the long run due to the dynamic coastal environment.
Doubleside
Double-sided equations can be solved by doing the same operation to both sides of the equals sign. This keeps the equation balanced. Terms should be combined so that variables are grouped together on one side and constants on the other. When a term crosses the equals sign to the other side, its sign changes. By following these steps, variables can be isolated so the equation can be solved.
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Module 4 topic 1 part 3 notes
- 1. Module 4 Topic 1 The Coordinate Plane Midpoint Formula Relations & Functions (Part 3)
- 8. In Figure 1, 3 is mapped to 5 and 6…thus making this NOT A FUNCTION !
- 9. In Figure 2, All values in the domain are mapped to different range values…thus making this A FUNCTION!
- 10. Are Figures 3 & 4 Function? Figure 3 is a Function! Figure 4 is NOT a Function!
- 15. Each graph to the right IS A FUNCTION! Notice the vertical line? It is only crossing the graphs once VERTICALLY!
- 16. Each graph to the right IS NOT A FUNCTION! Notice the vertical line? It is crossing each of the graphs more than once VERTICALLY!
- 17. You are now ready to begin the Module 4 Topic 1 Assignments… The Coordinate Plane Midpoint Formula Relations & Functions