A data table is shown below.
x	y
−6.1
−8.4
−5.9
−7.6
−4.5
−3.2
−3.1
−1.0
1.1
0
1.2
0.1
1.9
0.4
4.8
3.8
5.3
5.9
6.0
8.1

Which of the following function families best represents the data in the table?
 
A.Rational
B.Logarithmic
C.Exponential
D.Cubic

To determine which function family best represents the data in the table, let's analyze the general behavior of the pairs of \( x \) and \( y \) values and match them to the characteristics of the given function families.

Given the possible function families:

- **Rational:** Typically has asymptotes and may involve fractions, especially with the variable in the denominator.
- **Logarithmic:** Typically shows a curve that increases or decreases slowly, with a vertical asymptote.
- **Exponential:** Shows rapid increase or decrease, usually involving a constant base raised to the power of the variable.
- **Cubic:** Represents a polynomial of degree 3, which may involve changes in concavity and potentially multiple turning points.

### Analysis:
- The \( y \) values transition from negative to positive and seem to have a smooth curve rather than sharp asymptotes.
- There's a noticeable increase and decrease in the \( y \) values, which suggests that the function changes direction.
- Cubic functions have such characteristics, with a turning point and changes in concavity, which is consistent with the smooth rise and fall in the \( y \) values across the \( x \) range.

Given this analysis, **Cubic** is the best match because it can model data that increases, decreases, and changes direction smoothly.

**Answer: D. Cubic**

Would you like more details on how cubic functions behave or have any questions? Here are five related questions to consider:

1. How can you graphically identify a cubic function from data?
2. What are the key features of a rational function compared to a cubic function?
3. How do logarithmic functions differ from exponential functions in terms of growth rate?
4. Can a rational function ever resemble a cubic function? Why or why not?
5. What are the typical real-world applications for each of these function families?

**Tip:** When analyzing data, always consider the general shape and behavior of the graph when matching it to a function family.

Discover which function family (Rational, Logarithmic, Exponential, or Cubic) best represents a given dataset. This analysis focuses on understanding the behavior of the data points and matching them to characteristic curves of each function family. Suitable for high school students learning about different types of functions.

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