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Graph from slope-intercept form- Khan Academy assignment

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Graph from slope-intercept form- Khan Academy assignment

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Unit 2 Notes

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How can I use key features to describe relationships between different functions?

How do I graph linear functions and identify key
features?

How can a graph be interpreted in context of the
situation?

How does graphing a linear inequality compare to graphing a linear equation?

How are exponential functions used in real life
situations?

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Students will identify all key parts of functions and evaluate functions.

Students will interpret linear functions in context.

Students will graph linear functions and interpret their graphs from parent functions.

Students will calculate and interpret the average rate of change of a function over a specified interval.

Students will create linear functions from linear models, understanding the rate of change in the situation.

Students will compare two linear functions from linear models and interpret their solution.

Students will graph linear inequalities and interpret their solution on the coordinate plane and also in
context.

Students will graph exponential functions, understand key features: asymptote, vertical shift, and the relationship to parent functions.

Students will discover that geometric sequences are exponential functions.

Students will solve exponential growth and decay functions from real world situations.

Students will construct arithmetic and geometric sequences and write the rules.

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PRIORITY:
A.REI.12​ Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.

F.IF.2​ Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.

F.IF.6 ​Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.★

F.IF.7​ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.

F.BF.1​ Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F.LE.1​ Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate
per unit interval relative to another.

F.LE.2​ Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs (include
reading these from a table).

SUPPORTING:
A.REI.10​ Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

F.IF.1​ Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).

F.IF.3​ Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for .

F.IF.4​ For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive,
or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F.IF.5​ Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F.IF.9​ Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.

F.BF.2​ Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

F.BF.3​ Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from
their graphs and algebraic expressions for them.

F.LE.3​ Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F.LE.5​ Interpret the parameters in a linear or exponential function in terms of a context.

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Notation
Model
Notation
Investment
Principal
Interest
Rate
Depreciation (decay)
Per
Appreciation  (growth)

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Function
Linear
Value
Equation
Evaluate
Table
Input
Output
Domain
Range
Graph
Relationship
Constant Rate of Change (slope) Sets
Subsets
Integers
Increase
Decrease
Term
Rule
Constant Function
Common Ratio
Rate
Evaluate
Table
Input
Output
Graph
Relationship
Data
Increase
Decrease
Term
Rule
Constant Function
Geometric Sequence
Arithmetic Sequence
Graph
Relationship Data
Curve
Exponential Function
Exponent
Increase
Decrease
Growth Factor
Term
Finite Sequence
Infinite Sequence
Rule (function/equation)

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Linear vs. exponential growth: from data

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Linear vs. exponential growth: from data

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Given a table with values of two quantities over time, determine whether the quantities form grow linearly or exponentially.

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Explicit formulas for arithmetic sequences

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Explicit formulas for arithmetic sequences

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Solve problems such as: The first four terms in an arithmetic sequence are 12, 5, -2, and -9. Find an explicit formula for the sequence.

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Extend geometric sequences

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Introduction to geometric sequences

Introduction to geometric sequences

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Extend geometric sequences (practice) | Khan Academy

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- [Voiceover] The number of branches of an oak tree and a birch tree since 1950 are represented by the following tables. So for the oak tree we see when time equals zero has 34 branches. After three years it has 46 branches, so on and so forth. And in the birch tree they give us similar data. At the beginning has eight branches. In 10 years has 33 branches and they give us all of that. What I want to think about in this video is how should we model these? If we want to model these with functions and the choices we'll give ourselves, there are other options, but the choices we'll give ourselves in this video are linear, and linear versus exponential functions. Which of these are going to be better for modeling this data? So let's first look at the oak tree. A key to realization is, whenever I have a fixed increase in time, so each of these steps, this is plus three years. So it's a fixed increase in time. What happens to my number of branches? Is it going to be a fixed change, or roughly a fixed change in which case a linear model might be good, or is it going to be a change that's dependent on where we were? So what am I talking about? So 34 to 46, that is plus 12. 46 to 59 is plus 13. 59 to 70 is plus 11. 70 to 82 is plus 12. So at first you mention, well this isn't exactly a fixed change. These numbers they seem to average right around 12, but when you're looking at real world data you're never gonna get something that is exact. The models are just going to give us a good fit. Are going to give us a good approximation of the behavior of the number of branches over time. For me, this is pretty close to a constant 12 branches a year. So I would construct a linear model here. I would say here branches as a function of time. Let me be clear, this isn't 12 branches per year, this is 12 branches every three years. This is 13 branches over three years. This was 11 branches over three years. We're going to average 12 branches over three years. So the number of branches we have, we're gonna start at 34 branches and then minus. If we have 12 branches every three years, that's four branches every, or I should say plus. Four branches every year. And you could test this out. B of zero is gonna give us 34 branches. B of 12. Let's just really test out the extreme part of the model. B of 12 is going to be 34 plus 48, which is equal to 82. So this model works quite well. It's gonna have a couple of places where it's not exactly fitting the data, but it fits it quite, quite well. This is the linear model. So this one is linear. We also got the birch tree. So time equals zero, so fixed change in time. Alright, so we have a fixed change in time every time we are moving into the future, a decade. Let's see, our change in branches. We go from eight to 33. So what is that? That is plus 25 branches. Then we go from 33 to 128. Well that's way more than 25 branches. That's going to be what? Five less than 100. So that's gonna be plus 95 branches. So this clearly is not a linear model. And so let's think in terms of an exponential model. How much do we have to multiply to go from... Did I do that right? 128 minus, yeah, if it was 133 then it would be 100 and it's five less than that, okay. So now let's think about it in terms of an exponential model. In terms of an exponential model what do we have to multiply for each step? So if we have a constant step in time, what do we multiply for how much we increase our branches? So if we go to eight to 33, it's gonna be approximately four. It's gonna be a little bit more than four. 33 to 128, well that's gonna be a little bit less than four, but it's approximately four. 33 times four would be 132, so we're close. 128 to 512, that's exactly four, right? That's exactly 120 times four is 480 plus 32, yup, that is exactly four. So times four. And so it looks like we keep multiplying by four every 10 years that go by. So one way to think about it is we could say here, BFT, the branches of T, our initial condition, our initial state is eight and now we could say our common factor is four. But if we want T to be in years, well every 10 years we multiply by a factor of four. So T has to go to 10 before we increase the exponent to one, or has to go to 20 until this exponent becomes two. So eight times four to the T over 10 power seems like a pretty good model. And you could even verify this for yourself if you like. Try out what B of 30 is going to be. B of 30 would be eight times four. 30 divided by 10 to the third power and what is that going to be? That's going to be four to the third is 64. 64. Eight times 64 is, it's 480 plus 32. It is 512. So once again, this exponential model for this data does a pretty good job.

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Linear vs. exponential growth: from data (video) | Khan Academy

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Voiceover] The number of branches of an oak tree and a birch tree since 1950 are represented by the following tables. So for the oak tree we see when time equals zero has 34 branches. After three years it has 46 branches, so on and so forth.

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- [Voiceover] The temperature of a glass of warm water after it's put in a freezer is represented by the following table. So we have time in minutes and then we have the corresponding temperature at different times in minutes. Which model for C of t, the temperature of the glass of water t minutes after it's served, best fits the data? So pause the video and see which of these models best fit the data. Alright, now let's work through this together. So in order for it, we see these choices. Some of these choices are exponential models. Some of these are linear models. In order for a linear model to be a good description, when you have a fixed change in time, you should have a fixed change in temperature. If you're dealing with an exponential model, then as you have a fixed change in time, you should be changing by the same factor so the amount you change from say minute one to minute two or from minute two to minute three, it's not going to be the exact same amount but it should be the same factor of where you started. So let's think about this. So here, our change in time is two minutes. What is the absolute change in temperature? So our absolute change in temperature is negative what, 15.7. Negative 15.7 and what if we viewed it as a multiplication? So what do multiply 80 by to get 64.3? Well, I can get a calculator out for that. So 64.3 divided by 80 is equal to 0.8 I'll just say approximately .8 So we could multiply by .0.8 This is going to be approximate. So to get from 80 to 64.3 I could either subtract by 15.7 if I'm dealing with a linear model or I can multiply by 0.8 Now if I increase my time again by two, I'm going from minute two to minute four, so Delta t is equal to two, the absolute change here is what? This is going to be not 12. This is going to be, my brain isn't functioning optimally. This was 64.7 then this would be 12 but it's four less than that so it's 11.6. Negative 11.6 but if you look at it as multiplying it by a factor, what would you have to multiply by approximately? Let's get the calculator back out. So if I said 52.7 divided by 64.3. Divided by 64.3 is equal to, that's about .82. So times 0.82. So just by looking at this, I can keep going but it looks like for our give and change in time, my absolute change in the number is not even close to being the same. If this was 15.6, I'd be like okay maybe there's a little bit of error here. Data that you collect in the real world is never going to be perfect. These are models that try to get us close to describing the data. But over here we keep multiplying it by a factor of roughly .8, roughly .8. Now you might be tempted to immediately say, okay, well that means that C of t is going to be equal to our initial temperature, 80 times a common ratio of 0.8 to the number of minutes that passed by. Now this is very tempting and it would be the case if this was one minute and if this was two minutes but our change in temperature each time is our change in temperature each time is two minutes so what we really should say is this is one way to think about it is that it takes two minutes to have a 0.8 change or to be multiplied by 0.8. So the real way to describe this would be t over two. Every two minutes when t is zero, we'd be at 80. After two minutes, we would take 80 times 0.8, which is what we got over here. After four minutes, it would be 80 times 0.8 squared. In fact, let's just verify that we feel pretty good about this. So if we had something like this. So t and C of t. So when t is zero, C of t is 80. When t is, well let me just do the same data that we have here. When t is two, we have 80 times, two over two is one so it's 80 times 0.8, which is pretty close to what we have over here. When t is four, it would be 80 .0.8 squared which is pretty close to what we have right over here. I can just calculate it for you. If I have .8 squared times 80, 51.2 Getting pretty close. This is a pretty good approximation. Pretty good model. So I'm liking this model. This isn't exactly one of the choices so how do we manipulate this a little bit? Well, we can remind ourselves that this is the same thing as 80 times 0.8 to the one half and then that to the t power. And what's .8 to the one half? So .8. That's the same thing as taking the square root of .8 It's roughly .89 So this is approximately 80 times 0.89 to the t power. And if you look at all of these choices, this one is pretty close to this. This model best fits the data, especially of the choices going, this is pretty close to the model that I just thought about. Now another way of doing it that might have been a little bit simpler. I like to do it this way because even if I didn't have choices, we would have gotten to something reasonable. Another way to do it is to say okay, 80 is our initial state. All of these whether you're talking about exponential or linear models, start with 80 when t is equal to zero but it's clearly not a linear model because we're not changing by even roughly the same amount every time but it looks like every two minutes we're changing by a factor of .8 so we're going to have an exponential model so you say okay, it will be one of these two choices. Now this one down here you could rule out because we're not changing by a factor of 0.8 or 0.81 every minute. We're changing by a factor of 0.81 every two minutes so you could have ruled that one out and then you could have deduced to this right over here and you could have said, look, if I'm changing by a factor of .9 every minute then that would have been .81 every two minutes which is pretty close to what we're seeing here changing by a factor of about .8 or .81 every two minutes. So once again that's why we like that first choice.

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Linear vs. exponential growth: from data (example 2) (video) | Khan Academy

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Voiceover] The temperature of a glass of warm water after it's put in a freezer is represented by the following table. So we have time in minutes and then we have the corresponding temperature at different times in minutes. Which model for C of t, the temperature of the glass of water t minutes after it's served, best fits the data?

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- [Instructor] We are asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. So let's write it like this in a table. So if we have the term, just so we have things straight, and then we have the value, and then we have the value of the term. I'll do a nice little table here. So our first term we saw is 15. Our second term is nine. Our third term is three. I'm just really copying this down, but I'm making sure we associate it with the right term. And then our fourth term, our fourth term is negative three. And they wanna ask, they want us to figure out what the 100th term of this sequence is going to be. So let's see what's happening here, if we can discern some type of pattern. So we went from the first term to the second term, what happened? 15 to nine, looks like we went down by six. It's always good to think about just how much the numbers changed by. That's always the simplest type of pattern. So we went down by six, we subtracted six. Then to go from nine to three, well we subtracted six again. We subtracted six again. And then to go from three to negative three, well we, we subtracted six again. We subtracted six again. So it looks like every term, you subtract six. So the second term is going to be six less than the first term. The third term is going to be 12 minus from the first term, or six subtracted twice. So in the third term, you subtract a six twice. In the fourth term, you subtract six three times. So whatever term you're looking at, you subtract six one less than that many times. Let me write this down just so, notice when your first term, you have 15 and you don't subtract six at all, or you could say you subtract six zero times. So you could say this is 15 minus six times or let me write it better this way, minus zero times six. That's what that first term is right there. What's the second term? This is 15, it's just we just subtracted six once, or you could say minus one times six. Or you could say plus one times negative six, either way, we're subtracting the six once. Now what's happening here? This is 15, this is 15 minus two times negative six, or sorry minus two times six, minus two times six. We're subtracting a six twice. What's the fourth term? This is 15 minus, we're subtracting the six three times from the 15, so minus three times six. So if you see the pattern here, when our term, when we have our fourth term, we have the term minus one right there, the fourth term we have a three, the third term we have a two, the second term we have a one. So if we had the nth term, if we just had the nth term here, what's this going to be? It's going to be 15 minus, you see it's going to be n minus one right here, right when n is four, n minus one is three. When n is three, n minus one is two. When n is two, n minus one is one. When n is one, n minus one is zero. So we're going to have, this term right here is n minus one, so minus n minus one times six. So if you wanna figure out the 100th term of this sequence, I didn't even have to write it in this general term, you could just look at this pattern. It's going to be, and I'll do it in pink, the 100th term in our sequence, I'll continue our table down, is gonna be what? It's going to be 15 minus 100 minus one, which is 99, times six, right? I just followed the pattern. One, you had a zero here, two, you had a one here, three, you had a two here, 100, you're gonna have a 99 here. So let's just calculate what this is. What's 99 times six? So 99 times six, actually you could do this in your head. You could say that's going to be six less than 100 times six, which is 600, and six less is 594. But if you didn't wanna do it that way, you just do it the old-fashioned way. Six times nine is 54, carry the five. Nine times six or six times nine is 54. 54 plus five is 594. So this right here is 594, and then to figure out what 15, so we wanna figure out, we wanna figure out what 15 minus 594 is, and this can sometimes be confusing, but the way I always process this in my head is I say that this is the exact same thing as the negative of 594 minus 15. And if you don't believe me, distribute out this negative sign. Negative one times 594 is negative 594, negative one times negative 15 is positive 15. So these two statements are equivalent. This is much easier for my brain to understand. So what's 594 minus 15? You should do it, we could do this in our head. 594 minus 14 would be 580, and then 580 minus one more would be 579. So that right there is 579, and then we have this negative sign sitting out there. So our the 100th term in our sequence will be negative 579.

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Arithmetic sequence problem | Algebra (video) | Khan Academy

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Instructor] We are asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. So let's write it like this in a table.

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In this video I want to introduce you to the idea of a geometric sequence. And I have a ton of more advanced videos on the topic, but it's really a good place to start, just to understand what we're talking about when someone tells you a geometric sequence. Now a good starting point is just, what is a sequence? And a sequence is, you can imagine, just a progression of numbers. So for example, and this isn't even a geometric series, if I just said 1, 2, 3, 4, 5. This is a sequence of numbers. It's not a geometric sequence, but it is a sequence. A geometric sequence is a special progression, or a special sequence, of numbers, where each successive number is a fixed multiple of the number before it. Let me explain what I'm saying. So let's say my first number is 2 and then I multiply 2 by the number 3. So I multiply it by 3, I get 6. And then I multiply 6 times the number 3, and I get 18. Then I multiply 18 times the number 3, and I get 54. And I just keep going that way. So I just keep multiplying by the number 3. So I started, if we want to get some notation here, this is my first term. We'll call it a1 for my sequence. And each time I'm multiplying it by a common number, and that number is often called the common ratio. So in this case, a1 is equal to 2, and my common ratio is equal to 3. So if someone were to tell you, hey, you've got a geometric sequence. a1 is equal to 90 and your common ratio is equal to negative 1/3. That means that the first term of your sequence is 90. The second term is negative 1/3 times 90. Which is what? That's negative 30, right? 1/3 times 90 is 30, and then you put the negative number. Then the next number is going to be 1/3 times this. So negative 1/3 times this. 1/3 times 30 is 10. The negatives cancel out, so you get positive 10. Then the next number is going to be 10 times negative 1/3, or negative 10/3. And then the next number is going to be negative 10/3 times negative 1/3 so it's going to be positive 10/3. And you could just keep going on with this sequence. So that's what people talk about when they mean a geometric sequence. I want to make one little distinction here. This always used to confuse me because the terms are used very often in the same context. These are sequences. These are kind of a progression of numbers. 2, then 6, then 18, 90, then negative 30, then 10, then negative 10/3. Then, I'm sorry, this is positive 10/9, right? Negative 1/3 times negative 10/3, negatives cancel out. Right. 10/9. Don't want to make a mistake here. These are sequences. You might also see the word a series. And you might even see a geometric series. A series, the most conventional use of the word series, means a sum of a sequence. So for example, this is a geometric sequence. A geometric series would be 90 plus negative 30, plus 10, plus negative 10/3, plus 10/9. So a general way to view it is that a series is the sum of a sequence. I just want to make that clear because that used to confuse me a lot when I first learned about these things. But anyway, let's go back to the notion of a geometric sequence, and actually do a word problem that deals with one of these. So they're telling us that Anne goes bungee jumping off of a bridge above water. On the initial jump, the cord stretches by 120 feet. So on a1, our initial jump, the cord stretches by 120 feet. We could write it this way. We could write, jump, and then how much the cord stretches. So on the initial jump, on jump one, the cord stretches 120 feet. Then it says, on the next bounce, the stretch is 60% of the original jump, and then each additional bounce stretches the rope 60% of the previous stretch. So here, the common ratio, where each successive term in our sequence is going to be 60% of the previous term. Or it's going to be 0.6 times the previous term. So on the second jump, we're going to start 60% of that, or 0.6 times 120. Which is equal to what? That's equal to 72. Then on the third jump, we're going to stretch 0.6 of 72, or 0.6 times this. So it would be 0.6 times 0.6 times 120. Notice, over here, so on the fourth jump we're going to have 0.6 times 0.6 times 0.6 times 120. 60% of this jump, so every time we're 60% of the previous jump. So if we wanted to make a general formula for this, just based on the way we've defined it right here. So the stretch on the nth jump, what would it be? So let's see, we start at 120 times 0.6 to the what? To the n minus 1. How did I get this? Let me write this a little bit here. So this is equal to 0.6, actually let me write the 120 first. This is equal to 120 times 0.6 to the n minus 1. How did I get that? Well we're defining the first jump as stretching 120 feet. So when you put n is equal to 1 here, you get 1 minus 1, 0. So you have 0.6 to the 0th power, and you've just got a 1 here. And that's exactly what happened on the first jump. Then on the second jump, you put a 2 minus 1, and notice 2 minus 1 is the first power, and we have exactly one 0.6 here. So I figured it was n minus 1 because when n is 2, we have one 0.6, when n is 3, we have two 0.6's multiplied by themselves. When n is 4, we have 0.6 to the third power. So whatever n is, we're taking 0.6 to the n minus 1 power, and of course we're multiplying that times 120. Now and the question they also ask us, what will be the rope stretch on the 12th bounce? And over here I'm going to use the calculator. and actually let me correct this a little bit. This isn't incorrect, but they're talking about the bounce, and we could call the jump the zeroth bounce. Let me change that. This isn't wrong, but I think this is where they're going with the problem. So you can view the initial stretch as the zeroth bounce. So instead of labeling it jump, let me label it bounce. So the initial stretch is the zeroth bounce, then this would be the first bounce, the second bounce, the third bounce. And then our formula becomes a lot simpler. Because if you said the stretch on nth bounce, then the formula just becomes 0.6 to the n times 120, right? On the zeroth bounce, that was our original stretch, you get 0.6 to the 0, that's 1 times 120. On the first bounce, 0.6 to the 1, one 0.6 right here. 0.6 times the previous stretch, or the previous bounce. So this has it in terms of bounces, which I think is what the questioner wants us to do. So what about the 12th bounce? Using this convention right there. So if we do the 12th bounce, let's just get our calculator out. We're going to have 120 times 0.6 to the 12th power. And hopefully we'll get order of operations right, because exponents take precedence over multiplication, so it'll just take the 0.6 to the 12th power only. And so this is equal to 0.26 feet. So after your 12th bounce, she's going to be barely moving. She's going to be moving about 3 inches on that 12th bounce. Well, hopefully you found this helpful. And I apologize for the slight divergence here, but I actually think on some level that's instructive. Because you always have to make sure that your n matches well with what your results are. So when I talked about your first jump, I said, OK this is 1. And then I had 0.6 to the zeroth power, so I did n minus 1. But then when I relabeled things in terms of bounces, this was the zeroth bounce. This makes sense that this is 0.6 to the zeroth power. This is the first bounce, so this would be 0.6 to the first power. Second bounce, 0.6 to the second power. It made our equation a little bit simpler. Anyway, hopefully you found that Interesting.

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Intro to geometric sequences (video) | Khan Academy

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In this video I want to introduce you to the idea of a geometric sequence. And I have a ton of more advanced videos on the topic, but it's really a good place to start, just to understand what we're talking about when someone tells you a geometric sequence.

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Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value. For example, for any one-step change in x, is the change in y always going to be 3? Is it always going to be 5? If it's always going to be the same value, you're dealing with a linear function. If for each change in x--so over here x is always changing by 1, so since x is always changing by 1, the change in y's have to always be the same. If they're not, then we're dealing with a non-linear function. We can actually show that plotting out. If the changes in x-- we're going by different values, if this went from 1 to 2 and then 2 to 4-- what you'd want to do, then, is divide the change in y by the change in x, and that should always be a constant. In fact, let me write that down. If something is linear, then the change in y over the change in x always constant. Now, in this example, the change in x's are always 1, right? We go from 1 to 2, 2 to 3, 3 to 4, 4 to 5. So in this example, the change in x is always going to be 1. So in order for this function to be linear, our change in y needs to be constant because we're just going to take that and divide it by 1. So let's see if our change in y is constant. When we go from 11 to 14, we go up by 3. When we go from 14 to 19, we go up by 5, so I already see that it is not constant. We didn't go up by 3 this time, we went up by 5. And here, we go up by 7. And here, we're going up by 9. So we're actually going up by increasing amounts, so we're definitely dealing with a non-linear function. And we can see that if we graph it out. So let me draw-- I'll do a rough graph here. So let me make that my vertical axis, my y-axis. And we go all the way up to 35. So I'll just do 10, 20, 30. Actually, I can it do a little bit more granularly than that. I could do 5, 10, 15, 20, 25, 30, and then 35. And then our values go 1 through 5. I'll do it on this axis right here. They're not obviously the exact same scale, so I'll do 1, 2, 3, 4, and 5. So let's plot these points. So the first point is 1, 11, when x is 1, y is 11. This is our x-axis. When x is 1, y is 11, that's right about there. When x is 2, y is 14, that's right about there. When x is 3, y is 19, right about there. When x is 4, y is 26, right about there. And then finally, when x is 5, y is 35, right up there. So you can immediately see that this is not tracing out a line. If this was a linear function, then all the points would be on a line that looks something like that. That's why it's called a linear function. In this case, it's not, it's non-linear. The rate of increase as x changes is going up.

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Recognizing linear functions (video) | Khan Academy

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Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value.

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This right here is a picture of Rene Descartes. Once again, one of the great minds in both math and philosophy. And I think you're seeing a little bit of a trend here, that the great philosophers were also great mathematicians and vice versa. And he was somewhat of a contemporary of Galileo. He was 32 years younger, although he died shortly after Galileo died. This guy died at a much younger age. Galileo was well into his 70s. Descartes died at what is only 54 years old. And he's probably most known in popular culture for this quote right over here, a very philosophical quote. "I think, therefore I am." But I also wanted to throw in, and this isn't that related to algebra, but I just thought it was a really neat quote, probably his least famous quote, this one right over here. And I like it just because it's very practical, and it makes you realize that these great minds, these pillars of philosophy and mathematics, at the end of the day, they really were just human beings. And he said, "You just keep pushing. You just keep pushing. I made every mistake that could be made. But I just kept pushing." Which I think is very, very good life advice. Now, he did many things in philosophy and mathematics. But the reason why I'm including him here as we build our foundations of algebra is that he is the individual most responsible for a very strong connection between algebra and geometry. So on the left over here, you have the world of algebra, and we've discussed it a little bit. You have equations that deal with symbols, and these symbols are essentially they can take on values. So you could have something like y is equal to 2x minus 1. This gives us a relationship between whatever x is and whatever y is. And we can even set up a table here and pick values for x and see what the values of y would be. And I could just pick random values for x, and then figure out what y is. But I'll pick relatively straightforward values just so that the math doesn't get too complicated. So for example, if x is negative 2, then y is going to be 2 times negative 2 minus 1, which is negative 4 minus 1, which is negative 5. If x is negative 1, then y is going to be 2 times negative 1 minus 1, which is equal to-- this is negative 2 minus 1, which is negative 3. If x is equal to 0, then y is going to be 2 times 0 minus 1. 2 times 0 is 0 minus 1 is just negative 1. I'll do a couple more. And I could have picked any values here. I could have said, well, what happens if x is the negative square root of 2, or what happens if x is negative 5/2, or positive 6/7? But I'm just picking these numbers because it makes the math a lot easier when I try to figure out what y is going to be. But when x is 1, y is going to be 2 times 1 minus 1. 2 times 1 is 2 minus 1 is 1. And I'll do one more. I'll do one more in a color that I have not used yet. Let's see, this purple. If x is 2, then y is going to be 2 times 2-- now our x is 2-- minus 1. So that is 4 minus 1 is equal to 3. So fair enough. I just kind of sampled this relationship. I said, OK, this describes the general relationship between a variable y and a variable x. And then I just made it a little bit more concrete. I said, OK, well, then for each of these values of x, what would be the corresponding value of y? And what Descartes realized is is that you could visualize this. One, you could visualize these individual points, but that could also help you, in general, to visualize this relationship. And so what he essentially did is he bridged the worlds of this kind of very abstract, symbolic algebra and that and geometry, which was concerned with shapes and sizes and angles. So over here you have the world of geometry. And obviously, there are people in history, maybe many people, who history may have forgotten who might have dabbled in this. But before Descartes, it's generally viewed that geometry was Euclidean geometry, and that's essentially the geometry that you studied in a geometry class in eighth or ninth grade or 10th grade in a traditional high school curriculum. And that's the geometry of studying the relationships between triangles and their angles, and the relationships between circles and you have radii, and then you have triangles inscribed in circles, and all the rest. And we go into some depth in that in the geometry playlist. But Descartes said, well, I think I can represent this visually the same way that you could with studying these triangles and these circles. He said, well, if we view a piece of paper, if we think about a two-dimensional plane, you could view a piece of paper as kind of a section of a two-dimensional plane. And we call it two dimensions because there's two directions that you could go in. There's the up/down direction. That's one direction. So let me draw that. I'll do it in blue because we're starting to visualize things, so I'll do it in the geometry color. So you have the up/down direction. And you have the left/right direction. That's why it's called a two-dimensional plane. If we're dealing in three dimensions, you would have an in/out dimension. And it's very easy to do two dimensions on the screen because the screen is two dimensional. And he, says, well, you know, there are two variables here, and they have this relationship. So why don't I associate each of these variables with one of these dimensions over here? And by convention, let's make the y variable, which is really the dependent variable-- the way we did it, it depends on what x is-- let's put that on the vertical axis. And let's put our independent variable, the one where I just randomly picked values for it to see what y would become, let's put that on the horizontal axis. And it actually was Descartes who came up with the convention of using x's and y's, and we'll see later z's, in algebra so extensively as unknown variables or the variables that you're manipulating. But he says, well, if we think about it this way, if we number these dimensions-- so let's say that in the x direction, let's make this right over here is negative 3. Let's make this negative 2. This is negative 1. This is 0. Now, I'm just numbering the x direction, the left/right direction. Now this is positive 1. This is positive 2. This is positive 3. And we could do the same in the y direction. So let's see, so this could be, let's say this is negative 5, negative 4, negative 3, negative-- actually, let me do it a little bit neater than that. Let me clean this up a little bit. So let me erase this and extend this down a little bit so I can go all the way down to negative 5 without making it look too messy. So let's go all the way down here. And so we can number it. This is 1. This is 2. This is 3. And then this could be negative 1, negative 2. And these are all just conventions. It could have been labeled the other way. We could've decided to put the x there and the y there and make this the positive direction and make this the negative direction. But this is just the convention that people adopted starting with Descartes. Negative 2, negative 3, negative 4, and negative 5. And he says, well, I think I can associate each of these pairs of values with a point in two dimensions. I can take the x-coordinate, I can take the x value right over here, and I say, OK, that's a negative 2. That would be right over there along the left/right direction. I'm going to the left because it's negative. And that's associated with negative 5 in the vertical direction. So I say the y value is negative 5, and so if I go 2 to the left and 5 down, I get to this point right over there. So he says, these two values, negative 2 and negative 5, I can associate it with this point in this plane right over here, in this two-dimensional plane. So I'll say that point has the coordinates, tells me where to find that point, negative 2, negative 5. And these coordinates are called Cartesian coordinates, named for Rene Descartes because he's the guy that came up with these. He's associating, all of a sudden, these relationships with points on a coordinate plane. And then he said, well, OK, let's do another one. There's this other relationship, where I have when x is equal to negative 1, y is equal to negative 3. So x is negative 1, y is negative 3. That's that point right over there. And the convention is, once again, when you list the coordinates, you list the x-coordinate, then the y-coordinate. And that's just what people decided to do. Negative 1, negative 3, that would be that point right over there. And then you have the point when x is 0, y is negative 1. When x is 0 right over here, which means I don't go to the left or the right, y is negative 1, which means I go 1 down. So that's that point right over there, 0, negative 1, right over there. And I could keep doing this. When x is 1, y is 1. When x is 2, y is 3. Actually, let me do it in that same purple color. When x is 2, y is 3, 2 comma 3. And then this one right over here in orange was 1 comma 1. And this is neat by itself. I essentially just sampled possible x's. But what he realized is, not only do you sample these possible x's, but if you just kept sampling x's, if you tried sampling all the x's in between, you would actually end up plotting out a line. So if you were to do every possible x, you would end up getting a line that looks something like that right over there. And any relation, if you pick any x and find any y, it really represents a point on this line. Or another way to think about it, any point on this line represents a solution to this equation right over here. So if you have this point right over here, which looks like it's about x is 1 and 1/2, y is 2. So me write that, 1.5 comma 2. That is a solution to this equation. When x is 1.5, 2 times 1.5 is 3 minus 1 is 2. That is right over there. So all of a sudden, he was able to bridge this gap or this relationship between algebra and geometry. We can now visualize all of the x and y pairs that satisfy this equation right over here. And so he is responsible for making this bridge, and that's why the coordinates that we use to specify these points are called Cartesian coordinates. And as we'll see, the first type of equations we will study are equations of this form over here. And in a traditional algebra curriculum, they're called linear equations. And you might be saying, well, OK, this is an equation. I see that this is equal to that. But what's so linear about them? What makes them look like a line? And to realize why they are linear, you have to make this jump that Rene Descartes made, because if you were to plot this using Cartesian coordinates on a Euclidean plane, you will get a line. And in the future, we'll see that there's other types of equations where you won't get a line, where you'll get a curve or something kind of crazy or funky.

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Intro to the coordinate plane (video) | Khan Academy

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This right here is a picture of Rene Descartes. Once again, one of the great minds in both math and philosophy. And I think you're seeing a little bit of a trend here, that the great philosophers were also great mathematicians and vice versa. And he was somewhat of a contemporary of Galileo.

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Interpreting a graph example (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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We're asked to graph y is equal to 2.5 times x. So we really just have to think about two points that satisfy this equation here, and the most obvious one is what happens when x equals 0. When x equals 0, 2.5 times 0 is going to be 0. So when x is 0, y is going to be equal to 0. And then let's just pick another x that will give us a y that is a whole number. So if x increases by 1, y is going to increase by 2.5. It's going to go right over there, and I could graph it just like that. And we see just by what I just said that the unit rate of change of y with respect to x is 2.5. A unit increase in x, an increase of 1 and x, results in a 2.5 increase in y. You see that right over here. x goes from 0 to 1, and y goes from 0 to 2.5. But let's increase x by another 1, and then y is going to increase by 2.5 again to get to 5. Or you could say, hey, look, if x is equal to 2, 2.5 times 2 is equal to 5. So this is a legitimate graph for this equation, but then they also tell us to select the statements that are true. So the first one is the equation does not represent a proportional relationship. Well, this is a proportional relationship. A proportional relationship is one where, first of all, if you have zero x's, you're going to have zero y's, where y is equal to some constant times x. And here, y is equal to 2.5 times x. So this is definitely a proportional relationship, so I'm not going to check that. The unit rate of the relationship is 2/5. So I'm assuming-- this is a little ambiguous the way they stated it. I'm assuming they're saying the unit rate of change of y with respect to x. And the unit rate of change of y with respect to x is, when x increases 1, y changed 2.5. So here they're saying when x changes by 1, y changes by 0.4, 2/5 is the same thing as 0.4. This should be 5/2. 5/2 would be 2.4. So this isn't right as well. The slope of the line is 2.5. Well, this looks right. Slope is change in y over change in x. When x changes 1, y changes 2.5. So change in y, 2.5, over change in x, 1. 2.5 over 1 is 2.5. And you could also see it looking at the form of this equation. y is equal to-- this is the slope times x. So that's right. A change of 5 units in x results in a change of 2 units in y. Well, let's test that idea. We know when x is 0, y is 0. So if x goes from 0 to 5, what's going to happen to y? Well, y is going to be 2.5 times 5. 2.5 times 5 is 12.5. So y would not just change 2. It actually would change 12.5. So this isn't right. A change of 2 units in x results in a change of 5 units in y. Well, we see that. A change in 2 units of x results in a change of 5 units in y. That's exactly what we graphed right over here. These two points show that. So this is definitely true.

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Graphing proportional relationships from an equation (video) | Khan Academy

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We're asked to graph y is equal to 2.5 times x. So we really just have to think about two points that satisfy this equation here, and the most obvious one is what happens when x equals 0. When x equals 0, 2.5 times 0 is going to be 0.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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Slope of a line

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Interactive Slope, HTML5 Applet. Just Click, drag and Explore Graph and Formula of Slope

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Explore the slope formula with our interactive HTML5 applet. Shows rise, run and other features of the slope of a line

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Shodor > Interactivate > Activities > Slope Slider

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Interactive Slope, HTML5 Applet. Just Click, drag and Explore Graph and Formula of Slope

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Explore the slope formula with our interactive HTML5 applet. Shows rise, run and other features of the slope of a line

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Linear & nonlinear functions

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Determine if a relationship is linear or nonlinear.

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Interpreting graphs of functions

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Interpreting graphs of functions (practice) | Khan Academy

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Interpret the graphs of both linear and nonlinear functions.

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Graphing proportional relationships

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Graphing proportional relationships (practice) | Khan Academy

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If you're seeing this message, it means we're having trouble loading external resources on our website.

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Slope from equation

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Slope-intercept form (notes)

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Slope of a linear equation (notes)

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Linear or Non-Linear (notes)

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Linear or Non-linear (goes along with the notes)

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Graphing stories on Desmos

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Function (notes)

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Exponential Functions

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Linear vs. Exponential Functions

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Unit 2 final exam review- focusing on slope and linear equations

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Inequalities practice

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Graphing two-variable inequalities

Graphing two-variable inequalities

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Graphs of inequalities (practice) | Khan Academy

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Slope packet

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What's my rule? worksheet with video

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Am I Linear? worksheet

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Graphing Linear Inequalities

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Rate of change from a table