College Algebra & Trigonometry, 2018a

140 CHAPTER 3. EXPONENTS AND LOGARITHMS
The next type of mathematical relationshp is a multiplicative relationship. This
represents a situation in which one quantity is always a multiple of the other
quantity. This is commonly seen in proportional relationships. If a recipe for a
cake calls for 2 cups of flour, then, if we want to make 3 cakes, we’ll need 6 cups
of flour. The amount of flour (y) is always two times the number of cakes we
want to make (x): y =2x.
If a recipe for a batch of cookies (with 20 cookies per batch) calls for 1.5 cups
of sugar, then three batches would require 4.5 cups of sugar. The amount of
sugar required (y) is always the number of batches (x) times 1.5: y =1.5x. Ifwe
wanted to represent this relationship based on the number of cookies instead of
the number of batches, we would need to adjust the formula. Given that there are
20 cookies per batch, we could adjust our formula so that we first calculate the
number of batches from the number of cookies and then multiply by 1.5. If the
number of cookies is x and the amount of sugar is y, then y =1.5 ∗ x or y = 3 x.
20 40
The next type of mathematical relationship is the polynomial relationship. In this
type of relationship, one quantity is related to a power of another quantity. A
good example of this type of relationship involves gravity. As Galileo discovered
in the 16th century, the distance that an object falls after it is dropped is not
proportional to the time that it has been falling. Rather, it is proportional to the
square of the time. The table below shows this type of relationship.
t
d
1 16
2 64
3 144
4 256
After one second, it looks like the distance will always be sixteen times the time
the object has been falling. However, after two seconds, we can see that this
relationship no longer is true. That’s because this relationship is a polynomial
relationship in which the distance an object has fallen (d) is proportional to the
square of the time it has been falling (t): d =16t 2 .