To determine which function best fits the data shown in the scatterplot, we analyze the trend of the points:

1. The data points increase very rapidly as $x$ increases, indicating an exponential growth pattern rather than linear or quadratic growth.

Let’s examine each option:

1. $y = 100x + 2$: This is a linear function. The scatterplot shows rapid growth, not a straight-line trend, so this can be ruled out.
2. $y = 20x^2 + 10$: This is a quadratic function, where growth increases at a moderate rate ($x^2$). However, the scatterplot shows even faster growth, which excludes this choice.
3. $y = 4^x + 8$: This is an exponential function. Exponential functions grow extremely quickly as $x$ increases, which matches the behavior in the scatterplot.
4. $y = 2^x + 5$: This is also an exponential function, but it grows slower than $4^x$.

Comparison of Exponential Options:

The data shows extremely rapid growth as $x$ reaches values like 14–18. Exponential growth with a base of 4 (from $y = 4^x + 8$) fits this much better than $y = 2^x + 5$, as $4^x$ grows significantly faster than $2^x$.

Final Answer:

The function that best fits the data is: $\boxed{y = 4^x + 8}$

Would you like a step-by-step breakdown of exponential growth, or explanations for the other options?

Here are some related questions to expand your understanding:

1. How do you identify linear vs. quadratic vs. exponential patterns in data?
2. What are the characteristics of exponential growth compared to other types of growth?
3. How would the graph look for $y = 4^x + 8$ compared to $y = 2^x + 5$?
4. Why does the base of the exponential function affect the steepness of the graph?
5. How can you use a calculator to plot and compare exponential functions?

Tip: Exponential functions are often used to model population growth, compound interest, and other phenomena with rapid increases!