INTERPRETING LINEAR AND EXPONENTIAL FUNCTIONS ARISING IN APPLICATIONS
Mark's parents have decided to give Mark his allowance in a new way. They have decided that Mark will get twice the amount of money as the day before, with Mark's allowance per day starting at one penny every day he does his chores. Mark was calculative and extremely smart in Mathematics. He thought for a while and said yes to the proposal. Mark agreed to this becasue he saw that his money would always increase, and will never decrease. The domain of this function could be anything greater than 0 to positive infinity. However, in this situation, the domain would probably be from 1 to about 14, since the amount of money would already be $163.84. The amount of money he has will always be positive (since you cannot have an negative number of dollars!). So, as the number of days he does his chores increases, the amount of money he will get also increase. It will never be that as the number of days increases, the amount of money will decrease. This is a benefit of this type of allowance. So, the other end behavior of this situation would be as x approaches negative infinity, y approaches 0. The y-intercept of this function would be (0,1), since Mark starts out with one cent on the first day he does his chores. However, there would not be an x-intercept because the number of days Mark does his chores cannot be 0.
To represent the above situation we will construct the function using the formula:
y= a(b)^x
Where:
a is the y intercept
b is the common ratio
x is the amount of time
So, by using this formula, we can conclude that since in the situation, a=1 and b=2 the equation would be y=1(2)^x. From this equation, we can conclude the following:
x-intercept: none
y-intercept: (0,1)
Increasing: -infinity is less than x which is less than +infinity
Decreasing: none
Positive: This graph is always positive.
Negative: This graph is never negative.
End behaviors: As x -> ∞ , y -> ∞
As x -> - ∞ , y -> 0
To represent the above situation we will construct the function using the formula:
y= a(b)^x
Where:
a is the y intercept
b is the common ratio
x is the amount of time
So, by using this formula, we can conclude that since in the situation, a=1 and b=2 the equation would be y=1(2)^x. From this equation, we can conclude the following:
x-intercept: none
y-intercept: (0,1)
Increasing: -infinity is less than x which is less than +infinity
Decreasing: none
Positive: This graph is always positive.
Negative: This graph is never negative.
End behaviors: As x -> ∞ , y -> ∞
As x -> - ∞ , y -> 0
Graph of Y= 1(2)^x
The domain of this situation must always be all real positive numbers because money can never be in negative numbers. However, the rate of change of these domains might differ between different points.
Ex: Rate of change between (0,1) and (1,2) is 1.
On the other hand, the average rate of change for the points (2,4) and (3,8) is 4.
Hence, we can conclude that the rate of change between points in an exponential function is not the same between points. However, the common ratio(2) will always be the same.
Ex: Rate of change between (0,1) and (1,2) is 1.
On the other hand, the average rate of change for the points (2,4) and (3,8) is 4.
Hence, we can conclude that the rate of change between points in an exponential function is not the same between points. However, the common ratio(2) will always be the same.