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

wasa relationship betweenxandy, it might be a good idea to see if you can find the relationship. That is,y=f(x). We need to determine whatfis. In other words, what function, when we plug in the listed values ofx, gives us the corresponding values ofyfrom the table?The function, it turns out, is not that difficult to discover. When

xis 0,yis 1, and anxvalue of 3 yields 10 fory. It seems thatyis one more than the square ofxeach time. Therefore, we have the relationshipy=x^{2}+ 1You 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

xvalues 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

xvalues went in order, you might have looked for a pattern in theyvalues 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 nextyvalue.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

xvalues 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

xandy, 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

yincreasing or decreasing withxby the same amount for every equivalent change inx. Consider this table, provided courtesy of the Louisiana Department of Education (PDF):xyIn this table, when

xincreases by 1 from 5 to 6, theyvalue increases by 3 from 11 to 14. This is important information. Next, look at how much the values change whenxincreases by 4 from 6 to 10. Theyvalue here increases by 12, from 14 to 26.That means the relationship appears to be a

linearrelationship. Whenxincreased by 1,yincreased by 3, so whenxincreases by 4, which is 4 times 1, we would expectyto 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

xgoes from 10 to 14, an increase of 4, we would expect theyvalue to increase by 4 times 3, or 12. It does, sinceygoes 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= Ax+ Bwhere A and B are constants. If

ydecreases wheneverxincreases, A will be a negative number. In the case above,xandyboth 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 ofyas we did our exercise above. That value for our function table is 3. Thus we havey= 3x+ BThe value for B is whatever

yis whenxis 0. Since we are not given that value here, let’s not worry about it yet. We know whatyis whenxis 5, though, and that value is 11. We also know what happens toywhenxchanges by 1: it changes by 3 in the same direction as the change inx.With this information, we can figure out what B is in our rule: If

xchanges by -1 every timeychanges by -3, we simply need to subtract 15 from theyvalue atx= 5 to find theyvalue whenxis 0. Thus, 11 – 15 = -4, and B = -4.The rule, therefore, in this function table can be written as

y= 3x– 4Check 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.

xyFirst, verify whether the table shows a linear or nonlinear relationship. When

xincreases by 4 from 3 to 7,yincreases by 40 from 12 to 52. Next,xincreases again by 4 from 7 to 11. If the relationship is linear,ywill increase by the same amount as it did in the previous line. Butyincreases here from 52 to 124, a difference of 72, which is way more than 40. Therefore, we donothave a linear relationship in this table.Perhaps the relationship is quadratic, which would mean a square is involved. Quadratic equations have the form

y= Ax^{2}+ Bx+ Cwhere 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=x^{2}+ 3