Question

which of the following is the most appropriate equation to model the data?
$hat{y}=1.293(1.83)^x$
$hat{y}=0.83(1.293)^x$
$hat{y}=1.83x + 1.293$
$hat{y}=1.293x + 0.83$

Explanation:

Step1: Observe data pattern

The data points seem to follow an exponential - growth pattern rather than a linear pattern as the increase in the y - values is not constant for a unit increase in the x - values. So, we can rule out the linear equations $\hat{y}=1.83x + 1.293$ and $\hat{y}=1.293x + 0.83$.

Step2: Check exponential equations

For an exponential equation of the form $\hat{y}=a(b)^{x}$, when $x = 0$, $\hat{y}=a$. Looking at the graph, when $x = 0$, $y$ is approximately $1$. For the equation $\hat{y}=1.293(1.83)^{x}$, when $x = 0$, $\hat{y}=1.293$; for the equation $\hat{y}=0.83(1.293)^{x}$, when $x = 0$, $\hat{y}=0.83$. Also, by substituting some non - zero integer values of $x$ and comparing with the graph points, we find that $\hat{y}=1.293(1.83)^{x}$ better fits the data.

Answer:

$\hat{y}=1.293(1.83)^{x}$