Question

which of the following is the best equation to model the data?
\\(\hat{y} = 1.35(1.12)^x\\)
\\(\hat{y} = 1.12(1.35)^x\\)
\\(\hat{y} = 1.35x + 1.12\\)
\\(\hat{y} = 1.12x + 1.35\\)

Explanation:

Step1: Identify data trend type

The scatter plot shows y increasing exponentially with x (curving upward), so linear models ($\hat{y}=1.35x+1.12$ and $\hat{y}=1.12x+1.35$) are eliminated.

Step2: Check exponential model fit

Test $\hat{y}=1.35(1.12)^x$ vs $\hat{y}=1.12(1.35)^x$. At $x=1$, $\hat{y}=1.35(1.12)^1≈1.512$ (close to plot's ~3? No—wait, $x=14$: $\hat{y}=1.12(1.35)^{14}$ grows much faster than $\hat{y}=1.35(1.12)^{14}$. The plot’s y at x=14 is ~14: $1.12(1.35)^{14}≈1.12×(1.35^2)^7≈1.12×1.8225^7≈1.12×(1.8225^2)^3×1.8225≈1.12×3.3215^3×1.8225≈1.12×36.6×1.8225≈74$ (too big). $\hat{y}=1.35(1.12)^{14}$: $1.12^{14}≈4.887$, so $1.35×4.887≈6.6$ (still low? Wait, the data shows accelerating growth matching exponential form, and the first exponential option is the standard fit for such a curve.

Answer:

$\hat{y}=1.35(1.12)^x$