Much of Physics will involve identifying the pattern that exists between variables and using that pattern to predict the value that the independent variable has for any given change in the dependent variable. To make such predictions, you will need to reason proportionally. That is, you will need to understand the concept of doubling, tripling, quadrupling, halving, etc.  That's what this question is all about.
​

Becoming a Good Data Inspector
You have a lot to do in this question. Most of it involves inspecting a data set in order to find a pattern in the numbers. Doing this well will allow you to answer the first two parts of this three-part question correctly. Here's what you should do:


The Game Plan
Start with Row A. Read the X value for this row. Then look for another row that has an X value that is two times (or three times or four times depending on the version of the question that you have) the X-value of Row A.  When you find that row, we will call it the target row. Keep in mind the multiplier that was used to determine the target row's X value from the Row A X value.  Now compare the Y value for the target row to the Y value of Row A. Is it two times larger, four times larger, or one-half the size? That is, is the multiplier of the Y value the same multiplier, the inverse (or reciprocal) multiplier, or the square multiplier?

Once you've done the above, you can probably answer the first two parts of this question. You may need to know this about the three types of relationships:

• Linear Relationship: A doubling of X results in a doubling of Y. And a tripling of X results in a tripling of Y. Finally, a quadrupling of X results in a quadrupling of Y.
• Inverse Relationship:  If the X value changes by the multiplier M, then the Y value changes by the reciprocal multiplier - that is, 1/M. So doubling X causes Y to become one-half the value. Similarly, tripling X causes Y to become one-third the value. And so forth.
• Quadratic Relationship: If you observe that a doubling of X causes the Y value to increase by a factor of four, then you are observing a quadratic relationship. Similarly, a tripling of the X value could cause the Y value to be nine times greater. And a quadrupling of the X value could cause the Y value to be 16 times greater.  These are all indicators of a quadratic relationship.

Don't Hate. Extrapolate!
In the last part of this question, you have to extrapolate. That is, you have to determine the value of Y for a value of X that lies outside the domain of the given Data Set. The value of X is given; we will call it the target X value.  And it is either double, triple, or quadruple the value of X for one of the rows of the Data Table. Begin by finding the row in the Data Set that has an X value that is one-half, one-third, or one-fourth the target X value . Once you find it, remember what row it is; we will call this row the origin row.  (We're just making these terms up ... they have no standard meaning in the math-physics world).

Find the Y value of the origin row. Then use the pattern to extrapolate to determine the Y value that is paired with the target X value. This would be the answer. The Y value answer is either 4 times, 9 times, or 16 times the Y value of the origin row (depending on the version of the question that you have). Take your time; and be mentally organized as you approach this task. You got this!