Question
Functions: Linear vs Nonlinear
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00:0004:22
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00:02: We're gonna look at some different forms of relationships and we're gonna try and determine if they're linear or nonlinear. So in order to do so, we need to understand what linear is. So what is linear? Linear, by definition, is an equation of the form Y equals AX plus B, where you have one independent variable and one dependent variable, Y and X. The graph is a straight line. So it's kind of like a simple relationship between the two variables. So let's look at some different ways and let's determine if they're linear or nonlinear. 00:49: So here I have a table of X terms and Y terms, and I wanna determine if it represents a linear relationship or not. So I could do this a couple of different ways. I could plot the points and look at the graph or I could look at the change between each of them and make sure that it's a constant rate of change. So I'll actually do it both ways. So if I start with a graph... So let's see, I need to go 2, 4, 6, 8 and let's go 5, 10, 15, 20. So my first point is one, five. My second point is two, 10. My third point is four, 15. And my fourth point is eight, 20. So when I look at that, that does not look like a straight line. 01:40: Now, it could be that I just graphed it poorly. So let's double check it a different way. So what are my changes? So in each case, I can see that my Ys go up by five. And what do my Xs do? Here, my Xs went up by one. Here, they went up by two and here they went up by four. So if I look at the rate of change, the rate of change between the first two points is the change in Y is five over the change in X, which is one. Now, between the second two points, the rate of change in the Ys was still five, but the rate in change in the Xs is two, and right there I can stop. It does not have a constant rate of change or slope. So I know that this is not linear. 02:26: Let's look at another example. Here I have two scenarios, and we wanna read through each one of them. We wanna see if it determines a linear relationship. First one, for every one unit change in X, the Y is changed by four units. So if it's linear, then it has to have a slope, which is the change in Y over the change in X. So what does it tell us? That for every one unit change in the Xs, so my change in Xs is one, the Y is changed by four units. So I have a constant rate of change. No matter what, every time my Xs increase by one, my Ys are gonna increase by four. So this is linear. 03:10: Let's look at the second one. Your gas company charges you a base charge of $18 and then an additional $125 for every therm of gas you've used. So where are we seeing the change? We're seeing the change here, an additional $125 for every therm of gas you use. So this is gonna be a change in cost, which would be our change in Y, over a change in therm, which would be your change in X. So what do we know? That an additional $1.25, so our cost changes by $1.25 for every therm of gas, so for every one change in X. So this is linear. 04:00: Let's look at one more example. Here we have an equation. Is the function F of X equals 2X square plus seven linear? So remember, when we defined linear, we said linear is of the form Y equals AX plus B. What do you notice about X? That its exponent is one. Here the exponent is two. So right away, we know this is nonlinear.
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Question 1
Is the table linear?
Responses
A YesYes
B NoNo
Question 2
Is the function f(x) in the video linear?
Responses
A YesYes
B NoNo
Question 3
Are the gas company charges linear?
Responses
A YesYes
B NoNo
Video Player
00:0004:22
None
Captions
Show Transcript
00:02: We're gonna look at some different forms of relationships and we're gonna try and determine if they're linear or nonlinear. So in order to do so, we need to understand what linear is. So what is linear? Linear, by definition, is an equation of the form Y equals AX plus B, where you have one independent variable and one dependent variable, Y and X. The graph is a straight line. So it's kind of like a simple relationship between the two variables. So let's look at some different ways and let's determine if they're linear or nonlinear. 00:49: So here I have a table of X terms and Y terms, and I wanna determine if it represents a linear relationship or not. So I could do this a couple of different ways. I could plot the points and look at the graph or I could look at the change between each of them and make sure that it's a constant rate of change. So I'll actually do it both ways. So if I start with a graph... So let's see, I need to go 2, 4, 6, 8 and let's go 5, 10, 15, 20. So my first point is one, five. My second point is two, 10. My third point is four, 15. And my fourth point is eight, 20. So when I look at that, that does not look like a straight line. 01:40: Now, it could be that I just graphed it poorly. So let's double check it a different way. So what are my changes? So in each case, I can see that my Ys go up by five. And what do my Xs do? Here, my Xs went up by one. Here, they went up by two and here they went up by four. So if I look at the rate of change, the rate of change between the first two points is the change in Y is five over the change in X, which is one. Now, between the second two points, the rate of change in the Ys was still five, but the rate in change in the Xs is two, and right there I can stop. It does not have a constant rate of change or slope. So I know that this is not linear. 02:26: Let's look at another example. Here I have two scenarios, and we wanna read through each one of them. We wanna see if it determines a linear relationship. First one, for every one unit change in X, the Y is changed by four units. So if it's linear, then it has to have a slope, which is the change in Y over the change in X. So what does it tell us? That for every one unit change in the Xs, so my change in Xs is one, the Y is changed by four units. So I have a constant rate of change. No matter what, every time my Xs increase by one, my Ys are gonna increase by four. So this is linear. 03:10: Let's look at the second one. Your gas company charges you a base charge of $18 and then an additional $125 for every therm of gas you've used. So where are we seeing the change? We're seeing the change here, an additional $125 for every therm of gas you use. So this is gonna be a change in cost, which would be our change in Y, over a change in therm, which would be your change in X. So what do we know? That an additional $1.25, so our cost changes by $1.25 for every therm of gas, so for every one change in X. So this is linear. 04:00: Let's look at one more example. Here we have an equation. Is the function F of X equals 2X square plus seven linear? So remember, when we defined linear, we said linear is of the form Y equals AX plus B. What do you notice about X? That its exponent is one. Here the exponent is two. So right away, we know this is nonlinear.
Close
Question 1
Is the table linear?
Responses
A YesYes
B NoNo
Question 2
Is the function f(x) in the video linear?
Responses
A YesYes
B NoNo
Question 3
Are the gas company charges linear?
Responses
A YesYes
B NoNo
Answers
Answered by
GPT-4o mini
Based on the information provided in the video transcript:
**Question 1: Is the table linear?**
**Response:** B No
The analysis showed that the rate of change between the pairs of points was not constant, indicating that the relationship is nonlinear.
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**Question 2: Is the function f(x) in the video linear?**
**Response:** B No
The function given is \( f(x) = 2x^2 + 7 \), which has an exponent of 2 on \( x \), meaning it is not of the form \( Y = AX + B \) and is therefore nonlinear.
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**Question 3: Are the gas company charges linear?**
**Response:** A Yes
The video described a scenario where the charges increase by a constant amount ($125) for each therm of gas used, indicating a constant rate of change and thus a linear relationship.
**Question 1: Is the table linear?**
**Response:** B No
The analysis showed that the rate of change between the pairs of points was not constant, indicating that the relationship is nonlinear.
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**Question 2: Is the function f(x) in the video linear?**
**Response:** B No
The function given is \( f(x) = 2x^2 + 7 \), which has an exponent of 2 on \( x \), meaning it is not of the form \( Y = AX + B \) and is therefore nonlinear.
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**Question 3: Are the gas company charges linear?**
**Response:** A Yes
The video described a scenario where the charges increase by a constant amount ($125) for each therm of gas used, indicating a constant rate of change and thus a linear relationship.