The graph shown in the image appears to have a U-shaped curve, which suggests that it is likely modeled by a polynomial function with an even degree, such as a quadratic or quartic function. Based on the options provided:

1. $f(x) = 2^x$ – This is an exponential function, and it doesn’t match the U-shaped curve in the graph.
2. $f(x) = (-x + 1)^4$ – This is a quartic function but its symmetry would not match the one in the graph.
3. $f(x) = (x + 1)^3$ – This is a cubic function, and it does not fit the shape of the graph, which has a symmetry typically found in even-degree polynomials.
4. $f(x) = (x + 1)^4$ – This is a quartic function and it closely resembles the U-shape seen in the graph.

The best function that matches the data is:

$f(x) = (x + 1)^4$

This is a quartic function, and its U-shaped curve closely fits the points shown in the image.

Would you like more details on why this function fits better?

Here are five related questions to consider:

1. What is the general shape of even-degree polynomial functions?
2. How does a quartic function differ from a quadratic function?
3. How does shifting a function horizontally affect its graph?
4. What is the significance of the degree of a polynomial in determining its graph’s shape?
5. How can we determine the vertex or turning point of a polynomial function?

Tip: Higher-degree polynomials (like quartic) can exhibit more complex behavior, but they often maintain symmetry if they have only even exponents.