Friday, June 18, 2021

Function table pattern recognition (algebra I)

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The following question is in the public domain. It concerns the skill of analyzing a wide variety of patterns and functional relationships and/or extending those patterns and functional relationships, expressed numerically, algebraically, or geometrically. It was directly lifted from the web site of public release questions posted by a state department of education on the date of posting (link).

The table below shows a relationship between x and y.

  x     0     1     2     3     4     5  
y 1 2 5 10 ? ?

What are the next two values for y?

A. 11 and 13
B. 15 and 22
C. 16 and 25
D. 17 and 26

Paul Katulahttps://news.schoolsdo.org
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

3 COMMENTS

  1. The Web site says the correct answer is D (17 and 26), and I agree.

    As with most math problems, there are many ways to find the solution here. I will describe two of them.

    Since the problem stated that there was a relationship between x and y, it might be a good idea to see if you can find the relationship. That is, y = f (x). We need to determine what f is. In other words, what function, when we plug in the listed values of x, gives us the corresponding values of y from the table?

    The function, it turns out, is not that difficult to discover. When x is 0, y is 1, and an x value of 3 yields 10 for y. It seems that y is one more than the square of x each time. Therefore, we have the relationship

    y = x2 + 1

    You will find that this relationship works for all values in the table, and you should check it, just to make sure. It’s also not a good idea to use x values of 0 or 1 when trying to determine a relationship, since those numbers have lots of special properties.

    Anyway, we now have the simple task of plugging in x = 4 and 5, which give us, repectively, 4 x 4 + 1 = 17 and 5 x 5 + 1 = 26.

    Next, because you might have noticed that the x values went in order, you might have looked for a pattern in the y values listed in the table. This is a shortcut that should be used sparingly, since not all problems will have a relationship that is as simple as this.

    You notice that from 1 to 2 (the first two values of y), we add 1; from 2 to 5, we add 3; from 5 to 10, we add 5. So, we have added 1, then 3, then 5, to get the next y value.

    Logically, we should add 7 and then 9 to get the next two values, and that works as well. 10 + 7 = 17, and 17 + 9 = 26.

    As I said, be careful when using this approach, since it only works if the x values don’t skip around irregularly.

  2. A practice worksheet that uses a simpler form of the rule (no quadratic terms, just two operations of addition, subtraction, multiplication, and/or division) can be found here in a Adobe PDF file.

    This worksheet would be more appropriate for 6th graders, but if you are having difficulty figuring out what the rule is that describes the relationship between x and y, it might be a good idea to go back to the basics for a worksheet or two before moving on with more problems like this one.

  3. When trying to determine the pattern, first ask yourself if the relationship is linear. That is, a linear relationship will produce a function table that shows y increasing or decreasing with x by the same amount for every equivalent change in x. Consider this table, provided courtesy of the Louisiana Department of Education (PDF):

      x     5     6     10     14  
    y 11 14 26 38

    In this table, when x increases by 1 from 5 to 6, the y value increases by 3 from 11 to 14. This is important information. Next, look at how much the values change when x increases by 4 from 6 to 10. The y value here increases by 12, from 14 to 26.

    That means the relationship appears to be a linear relationship. When x increased by 1, y increased by 3, so when x increases by 4, which is 4 times 1, we would expect y to increase by 4 times 3, which is 12, if the relationship is linear. Since it did increase by that amount, we think we have a linear relationship.

    It is always safe to try one more, since the table has additional values. When x goes from 10 to 14, an increase of 4, we would expect the y value to increase by 4 times 3, or 12. It does, since y goes from 26 to 38 in this case.

    Therefore, our conclusion is that the function is linear in x, which means it will look something like this:

    y = A x + B

    where A and B are constants. If y decreases whenever x increases, A will be a negative number. In the case above, x and y both moved in the same direction, so A is a positive number in our relationship. As you might have guessed, it’s whatever number you multiply by to get the next value of y as we did our exercise above. That value for our function table is 3. Thus we have

    y = 3 x + B

    The value for B is whatever y is when x is 0. Since we are not given that value here, let’s not worry about it yet. We know what y is when x is 5, though, and that value is 11. We also know what happens to y when x changes by 1: it changes by 3 in the same direction as the change in x.

    With this information, we can figure out what B is in our rule: If x changes by -1 every time y changes by -3, we simply need to subtract 15 from the y value at x = 5 to find the y value when x is 0. Thus, 11 – 15 = -4, and B = -4.

    The rule, therefore, in this function table can be written as

    y = 3 x – 4

    Check it a few times to make sure: 3 x 6 – 4 = 14. Check. 3 x 10 – 4 = 26. Check.

    Another possibility, though, is that the table will not be a linear relationship, as we had in the original example. Let’s consider another table from the same source at the Louisiana Department of Education.

      x     2     3     7     11     13  
    y 7 12 52 124 172

    First, verify whether the table shows a linear or nonlinear relationship. When x increases by 4 from 3 to 7, y increases by 40 from 12 to 52. Next, x increases again by 4 from 7 to 11. If the relationship is linear, y will increase by the same amount as it did in the previous line. But y increases here from 52 to 124, a difference of 72, which is way more than 40. Therefore, we do not have a linear relationship in this table.

    Perhaps the relationship is quadratic, which would mean a square is involved. Quadratic equations have the form

    y = A x2 + B x + C

    where A, B, and C are constants or just plain numbers. Since we don’t know yet, let’s see if it’s even possible that the function table describes a quadratic relationship.

    We can ask ourself, mathematically, is there a value of A, B, and C for which A(11)^2 + B(11) + C = 124 and A(7)^2 + B(7) + C = 52? And just from noticing, if B = 0, we have 121 A + C = 124 and 49 A + C = 52. If A = 1, then C = 3 would make both of these true.

    121 A + C = 124
    – [49 A + C = 52]
    ——————
    72 A = 72, so A = 1. Then plug that value for A back into either equation, and we see that C = 3:

    121 (1) + C = 124, so C = 3.

    Thus, the rule for this function table, with A = 1, B = 0, and C = 3, is

    y = x2 + 3

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