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=11.5(0.96)^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
-20
-13
-6
1
8
15
y
25
19
14
11
8
6
Let's calculate the difference between consecutive y-values.
25-19&= 6, 19-14= 5,
14-11&=3, 11-8=3,
8-6&=2
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.
19/25&= 0.76, 14/19 ≈ 0.737, [0.8em]
11/14 &≈ 0.786 , 8/11 ≈ 0.727, [0.8em]
6/8&=0.75
Each ratio is around 0.75, 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. For simplicity, we will use (1,11) and (8,8). Let's start by substituting 1 for x and 11 for y.
Now that we know that a=11b, we can partially write the equation.
y=11/bb^x
To find the value of b, we will substitute 8 for x and 8 for y in the above equation.
