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
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.
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.
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):
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.
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