If the ratios of consecutive y-values are equal, then the data can be modeled by an exponential function. If the difference of consecutive y-values is constant, then the data can be modeled by a linear function.
Exponential relationship of data: Yes. Example model: y=10.23(1.20)^x Explanation: See solution.
Practice makes perfect
We want to determine whether the data show an exponential relationship. Then we will write a function that models the data. Let's do those things one at a time.
Determining the Type of the Model
If the ratios of consecutive y-values are equal, then the data can be modeled by an exponential function. If the difference of consecutive y-values is constant, then the data can be modeled by a linear function. Consider the given table.
x
1
6
11
16
21
y
12
28
76
190
450
Let's calculate the difference between consecutive y-values.
28-12&= 16, 76-28= 48,
190-76&=114, 450-190=260
We can see that the differences are not constant, so the data cannot be modeled by a linear function. Let's determine the ratios of the consecutive y-values.
12/28 &≈ 0.429, 28/76 ≈ 0.368, [0.8em]
76/190&=0.4 , 190/450 ≈ 0.422
Each ratio is around 0.4, so the data can be modeled by an exponential function.
y=ab^x
Writing the Model
To find the values of a and b, we will use two of the ordered pairs given in the table. We will use (11,76) and (16,190). Let's start by substituting 11 for x and 76 for y.
Now that we know that a=76b^(11), we can partially write the equation.
y=76/b^(11)b^x
To find the value of b, we will substitute 16 for x and 190 for y in the above equation.
