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Recognizing linear functions

Learn to recognize if a function is linear. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pants teal style avatar for user Ingrid
    So the non-linear function in this video is a parabola?
    (0 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      In Sal's table, notice that every value of y equals 10 plus x^2
      When x = 1, y = (1)² + 10 = 11
      When x = 2, y = (2)² + 10 = 14
      When x = 3, y = (3)² + 10 = 19
      When x = 4, y = (4)² + 10 = 26
      When x = 5, y = (5)² + 10 = 35
      So, in each case shown in the table, y = x² + 10 and that is definitely a quadratic. A quadratic describes the points that make a parabola.

      Technically, though, we don't know if this function is continuous or if it is defined by that table and only has those 5 points. Sal only said that the function contains those points and no one tells us that there are any other points in the function. We haven't been told if x = 0 is included or x = 1/2 or x = -3

      Anyway, those points in the table do lie on a parabola--we just don't know if there are any points between those. If the problem said that the function was defined by
      y = x² + 10, or if it showed the curve of a parabola with those points on it, then we would know that all the points were included. . . but then the video wouldn't be making Sal's point which is how you can know that a function is linear just by looking at the table and this one is definitely not linear.
      (42 votes)
  • purple pi purple style avatar for user Alex
    So would a function with the following points be a linear function?
    (1,1)
    (2,4)
    (4,7)
    (8,10)
    (16,13)
    (32,16)
    The change in x is constant, it's always x times 2.
    The change in y is constant, it's always y plus 3.
    But when these points are plotted on a graph, there is no straight line between them?
    How can you have a constant change in x and y but a nonlinear function?
    (10 votes)
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    • orange juice squid orange style avatar for user Shayaan Qureshi
      well, you are not having a constant change in x and y.
      To go from x = 1 to x = 2, you add 1. to go from y = 1 to y = 4, you add 3. it's okay for now. But to go from x = 2 to x = 4, you add 2, so you should add 3*2 =6 to the previous y (i.e.,4) to get 10, but you added only 3 to get 7.
      (1 vote)
  • leaf blue style avatar for user William Lopez
    how do I know if a function is linear or not when it is explained like this: f(x)=x-11; (4) ?
    (8 votes)
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  • starky sapling style avatar for user kbui
    Would something like y=3 be linear or nonlinear?
    (4 votes)
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  • marcimus pink style avatar for user Falana Evelyn-Moe, Lashavia
    I would ask for help but are people still on here to ask?
    (4 votes)
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  • starky seedling style avatar for user Mya Alexander
    So can negative number also be linear or is that just for positive numbers
    (4 votes)
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    • primosaur ultimate style avatar for user Jonathan Haerr
      Linear equations can have negative values in them! For example:
      x y
      -2 -5
      -1 -3
      0 -1
      1 1
      This set of values is linear, because every time x increases by 1, y goes up 2 so there is the same interval between each y value. This works even though there are negative numbers!
      (5 votes)
  • male robot donald style avatar for user Sanjay
    I still don't get what a linear functions is?
    (3 votes)
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    • duskpin tree style avatar for user Dan Donnelly
      A linear function refers to when the dependent variable (usually expressed by 'y') changes by a constant amount as the independent variable (usually 'x') also changes by a constant amount. For example, the number of times the second hand on a clock ticks over time, is a linear function. Every minute (the constant change in x) the second hand ticks 60 times (the constant change in y). This is a linear function because for every 1 minute, the clock ticks the same number of times. If we express this situation on a graph, we would observe a straight diagonal ray, starting at (0,0) and increasing towards the upper right. As x (minutes) increases by 1, y (number of ticks) would increase by 60.
      (5 votes)
  • hopper happy style avatar for user miriam johnson
    i dont understand like the x and y dont dont agree with there constants
    (5 votes)
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  • leaf red style avatar for user Hosam Al Madani
    At you talk about seeing if it's Linear by dividing the change in Y by the in change X. I did not understand that?
    (2 votes)
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  • male robot hal style avatar for user Universe14
    How can you tell if the chart is increasing or decreasing or, do you just look at the y-value to see if the chart is increasing or decreasing.
    (4 votes)
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Video transcript

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.