To determine the best type of function to model the given data, we can analyze and compare the possible forms: linear, quadratic, and exponential.
Given data points:
- [tex]\( x = 1 \)[/tex], [tex]\( y = 9.68 \)[/tex]
- [tex]\( x = 2 \)[/tex], [tex]\( y = 19.36 \)[/tex]
- [tex]\( x = 3 \)[/tex], [tex]\( y = 38.72 \)[/tex]
- [tex]\( x = 4 \)[/tex], [tex]\( y = 77.44 \)[/tex]
- [tex]\( x = 5 \)[/tex], [tex]\( y = 154.88 \)[/tex]
### Evaluation of Model Types
Linear Model [tex]\( y = mx + b \)[/tex]:
- Linear models assume a constant rate of change (addition).
- We can observe whether [tex]\( y \)[/tex] increases by a constant amount as [tex]\( x \)[/tex] increases.
Checking differences:
[tex]\[ 19.36 - 9.68 = 9.68 \][/tex]
[tex]\[ 38.72 - 19.36 = 19.36 \][/tex]
[tex]\[ 77.44 - 38.72 = 38.72 \][/tex]
[tex]\[ 154.88 - 77.44 = 77.44 \][/tex]
The differences are not constant, so a linear model is not suitable.
Quadratic Model [tex]\( y = ax^2 \)[/tex]:
- Quadratic models assume the rate of change increases at a constant rate (squared relation).
- We can see if the change in [tex]\( y \)[/tex] relates to [tex]\( x^2 \)[/tex].
However, testing the provided data does not immediately suggest a consistent squared relationship without performing more complex regression, which indicates a simpler model might fit better.
Exponential Model [tex]\( y = a \cdot b^x \)[/tex]:
- Exponential models assume a multiplicative rate of change.
- We can check for a pattern of multiplication between consecutive [tex]\( y \)[/tex]-values.
Checking ratios:
[tex]\[ \frac{19.36}{9.68} = 2 \][/tex]
[tex]\[ \frac{38.72}{19.36} = 2 \][/tex]
[tex]\[ \frac{77.44}{38.72} = 2 \][/tex]
[tex]\[ \frac{154.88}{77.44} = 2 \][/tex]
There is a constant ratio of 2, indicating exponential growth.
### Exponential Model Determination
Given this constant ratio, we suspect an exponential model of the form [tex]\( y = a \cdot b^x \)[/tex]. The ratios suggest [tex]\( b = 2 \)[/tex].
Using the first data point ([tex]\( x = 1 \)[/tex], [tex]\( y = 9.68 \)[/tex]):
[tex]\[ 9.68 = a \cdot 2^1 \][/tex]
[tex]\[ 9.68 = 2a \][/tex]
[tex]\[ a = \frac{9.68}{2} = 4.84 \][/tex]
Thus, the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[ a = 4.84 \][/tex]
[tex]\[ b = 2 \][/tex]
Therefore, the exponential function that models the data is:
[tex]\[ y = 4.84 \cdot 2^x \][/tex]
