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=12.95(1.85)^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
-3
-1
1
3
5
y
2
7
24
68
194
Let's calculate the difference between consecutive y-values.
7-2&= 5, 24-7= 17,
68-24&=44, 194-68=126
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.
2/7 &≈ 0.286, 7/24 ≈ 0.292, [0.8em]
24/68 &≈ 0.353 , 68/194 ≈ 0.351
Each ratio is around 0.3, 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 (-1,7) and (1,24). Let's start by substituting -1 for x and 7 for y.
Now that we know that a=7b, we can partially write the equation.
y=7b(b)^x
To find the value of b, we will substitute 1 for x and 24 for y in the above equation.
