If you studied the writing equations unit, you learned how to write equations given two points and given slope and a point. We are going to use this same skill when working with functions.
The only thing different is the function notation. You first must be able to identify an ordered pair that is written in function notation. This can be a little tricky, but hopefully when you see this example, it will all make sense.
Ok, that was pretty easy, right? You already knew this skill, but it's coming back in a different format. Next we are going to take it one step further and find the slope of the graph for a linear function. Take a look at this example.
Is it all coming back to you now? Remember that in this particular function lesson, you really aren't learning any new material. You are applying what you know about equations and simply stating your answer in a much fancier format. We will continue studying linear functions in the next lesson, as we have a lot to cover. Keep going, you are doing great! Click here for more information on our affordable subscription options.
Not ready to subscribe? Click here for more information on our Algebra Class e-courses. On this site, I recommend only two products that I use and love. One is Mathway and the other is Magoosh. If you make a purchase on one of these sites, I may receive a small commission at no cost to you. Algebra Class. Linear Functions If you studied the writing equations unit, you learned how to write equations given two points and given slope and a point.
Linear Functions and Function Notation. Comments We would love to hear what you have to say about this page! Need Help? Try This Online Calculator! Affiliate Products Let Us Know How we are doing!If you're seeing this message, it means we're having trouble loading external resources on our website.
5 1 Understanding Linar Functions
Donate Login Sign up Search for courses, skills, and videos. Skill Summary Legend Opens a modal. Topic A: Functions. What is a function? Opens a modal. Worked example: Evaluating functions from equation Opens a modal. Worked example: Evaluating functions from graph Opens a modal. Equations vs. Manipulating formulas: temperature Opens a modal.
Testing if a relationship is a function Opens a modal. Relations and functions Opens a modal. Recognizing functions from graph Opens a modal. Checking if a table represents a function Opens a modal. Recognizing functions from verbal description Opens a modal.
Recognizing functions from table Opens a modal.
Recognizing functions from verbal description word problem Opens a modal. Checking if an equation represents a function Opens a modal. Does a vertical line represent a function? Evaluate functions Get 3 of 4 questions to level up!
Evaluate functions from their graph Get 3 of 4 questions to level up! Function rules from equations Get 3 of 4 questions to level up! Recognize functions from tables Get 3 of 4 questions to level up! Recognize functions from graphs Get 3 of 4 questions to level up!
Quiz 1. Topic B: Volume. Volume of a sphere Opens a modal. Volume of a cone Opens a modal. Volume of cylinders Get 3 of 4 questions to level up! Volume of spheres Get 3 of 4 questions to level up!English Language Arts. Students compare proportional relationships, define and identify slope from various representations, graph linear equations in the coordinate plane, and write equations for linear relationships.
In Unit 5, eighth-grade students zoom into linear functions, extending several ideas they learned in the previous unit on Functions.
They begin the unit by investigating and comparing proportional relationships, bridging concepts from seventh grade, such as constant of proportionality and unit rate, to new ideas in eighth grade, such as slope.
Students formally define slope and learn how to identify slope in various representations including graphs, tables, equations, and coordinate points. Investigating slope is an opportunity for students to engage in MP. Just as in Unit 4, students will draw on previous understandings from sixth and seventh grades related to rates and proportional relationships, and the equations and graphs that represent these relationships. The concepts and skills students learn in this unit are foundational to the next unit on systems of linear equations.
In Unit 6, students will investigate what happens when two linear equations are considered simultaneously. In high school, students will continue to build on their understanding of linear relationships and extend this understanding to graphing solutions to linear inequalities as half-planes in the coordinate plane.
For guidance on adjusting the pacing for the school year due to school closures, see our 8th Grade Scope and Sequence Recommended Adjustments. This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit. Write equations into slope-intercept form in order to graph.
Graph vertical and horizontal lines. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Expressions and Equations 8. For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Functions 8. Determine the rate of change and initial value of the function from a description of a relationship or from two x, y values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Here is the teacher edition, student edition and the videos associated with this lesson.
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Solutions to 2-variable equations Opens a modal. Worked example: solutions to 2-variable equations Opens a modal. Completing solutions to 2-variable equations Opens a modal. Solutions to 2-variable equations: substitution old Opens a modal. Solutions to 2-variable equations: graphical old Opens a modal. Solutions to 2-variable equations.
Complete solutions to 2-variable equations. Intro to intercepts Opens a modal. Intercepts from an equation Opens a modal. Intercepts from a table Opens a modal.
Graphing using intercepts old Opens a modal. Intercepts of lines review x-intercepts and y-intercepts Opens a modal. Intercepts from a graph. Intercepts from an equation.
Intercepts from a table. Intro to slope Opens a modal. Worked example: slope from graph Opens a modal. Worked example: slope from two points Opens a modal. Slope more examples Opens a modal.
Slope review Opens a modal. Slope from graph. Slope from two points. Slope of a horizontal line Opens a modal. Intro to slope-intercept form. Intro to slope-intercept form Opens a modal. Worked examples: slope-intercept intro Opens a modal.In the previous lesson on functions you learned how to find the slope and write an equation when given a function. Linear functions are very much like linear equations, the only difference is you are using function notation "f x " instead of "y".
Otherwise, the process is the same. Ok, let's move on! In our first example, we are going to find the value of x when given a value for f x. This is one of the trickier problems in the function unit.
Watch carefully where we substitute the given number 4. Pretty easy, right? This is really just a review of concepts that you've already learned. Once you figure out that you substitute 4 for f xyou solve this as a regular two step equation. This process for this problem is exactly the same as you learned when writing equations.
The only difference is how you state your "function" at the end. It must be written in function notation. This completes our lesson on Linear Functions. Hopefully you do not let the word "function" intimidate you. As you can see, you know how to solve all of these problems from studying equations. Now you just have to be a little fancier with how you name your equation. In the next lesson, we will continue our study of functions by taking a look at quadratic functions. Click here for more information on our affordable subscription options.
Not ready to subscribe? Click here for more information on our Algebra Class e-courses. On this site, I recommend only two products that I use and love.
One is Mathway and the other is Magoosh. If you make a purchase on one of these sites, I may receive a small commission at no cost to you. Algebra Class. Solving a Linear Function - Part 2 In the previous lesson on functions you learned how to find the slope and write an equation when given a function.
Let's take a look. Example 1: Solving for x in a linear function. Comments We would love to hear what you have to say about this page!How this works: This forecast is based on 100,000 simulations of the season and updates after every game. EasternElo point spreadWin prob. Score Green Bay -8. Score Kansas City -3. Score New Orleans -1. Score Kansas City -7. Score Green Bay -1. Score New Orleans -0. Score New England -6. Score Kansas City -0.
American author, inventor and futurist Raymond Kurzweil has become well known for his predictions about artificial intelligence and the human species, mainly concerning the technological singularity. He predicts that Artificial Intelligence would outsmart the human brain in computational capabilities by mid-21st century. His first book, The Age of Intelligent Machines, published in 1990, put forth his theories on the results of the increasing use of technology and predicted the explosive growth in the internet, among other predictions.
Later works, 1999's The Age of Spiritual Machines and 2005's The Singularity is Near outlined other theories including the rise of clouds of nano-robots (nanobots) called foglets and the development of Human Body 2.
Kurzweil's first book, The Age of Intelligent Machines was published in 1990. It forecast the demise of the Soviet Union due to new technologies such as cellular phones and fax machines disempowering authoritarian governments by removing state control over the flow of information.
He also stated that the Internet would explode not only in the number of users but in content as well, eventually granting users access "to international networks of libraries, data bases, and information services". The third and final section of the book is devoted to elucidating the specific course of technological advancements Kurzweil believes the world will experience over the next century. Titled "To Face the Future", the section is divided into four chapters respectively named "2009", "2019", "2029", and "2099".
For every chapter, Kurzweil issues predictions about what life and technology will be like in that year. The device was portable, but not the cheap, pocket-sized device of the prediction. While this book focuses on the future of technology and the human race as The Age of Intelligent Machines and The Age of Spiritual Machines did, Kurzweil makes very few concrete, short-term predictions in The Singularity Is Near, though longer-term visions abound.
Kurzweil predicted that, in 2005, supercomputers with the computational capacities to simulate protein folding will be introduced. In 2010, a supercomputer simulated protein folding for a very small protein at an atomic level over a period of a millisecond.
The protein folded and unfolded, with the results closely matching experimental data. Chess Champion and International Grandmaster Larry Christiansen in a four-game match.
Another 3 are partially correct, 2 look like they are about 10 years off, and 1, which was tongue in cheek anyway, was just wrong. Kurzweil said in a 2006 C-SPAN2 interview that "nanotechnology-based" flying cars would be available in 20 years. Kurzweil believes, by the end of the 2020s, humans will be able to completely replace fossil fuels. In the cover article of the December 2010 issue of IEEE Spectrum, John Rennie criticized Kurzweil's predictions: "On close examination, his clearest and most successful predictions often lack originality or profundity.