12.

Question

Which kind of function best models the data in the table? Graph the data and write an equation to model the data.

x y
0 –1
1 –2
2 –3
3 –4
4 –5


linear; y = –x – 1

exponential; y = 3x – 1

linear; y = x – 1

quadratic; y = x2 – 1

Answer 1

To determine which kind of function best models the given data, we can start by examining the relationship between \(x\) and \(y\). Given the data points:

\[
\begin{array}{cc}
x & y \\
0 & -1 \\
1 & -2 \\
2 & -3 \\
3 & -4 \\
4 & -5 \\
\end{array}
\]

We can see that as \(x\) increases by 1 unit, \(y\) decreases by 1 unit. This suggests a linear relationship.

### Checking Linear Functions

1. **Linear Function: \(y = -x - 1\)**

If we check \(y = -x - 1\):

- For \(x = 0\), \(y = -0 - 1 = -1\)
- For \(x = 1\), \(y = -1 - 1 = -2\)
- For \(x = 2\), \(y = -2 - 1 = -3\)
- For \(x = 3\), \(y = -3 - 1 = -4\)
- For \(x = 4\), \(y = -4 - 1 = -5\)

This equation perfectly fits all the given data points.

2. **Linear Function: \(y = x - 1\)**

If we check \(y = x - 1\):

- For \(x = 0\), \(y = 0 - 1 = -1\)
- For \(x = 1\), \(y = 1 - 1 = 0\)
- For \(x = 2\), \(y = 2 - 1 = 1\)
- For \(x = 3\), \(y = 3 - 1 = 2\)
- For \(x = 4\), \(y = 4 - 1 = 3\)

This does not fit the data points at all.

### Checking Other Functions

- **Exponential Function: \(y = 3^x - 1\)**

An exponential function does not fit the data because the rate of change here is constant (linear), and exponential functions have multiplicative rates of change rather than additive.

- **Quadratic Function: \(y = x^2 - 1\)**

Quadratic functions also do not fit the data because they involve squared terms and exhibit a parabolic shape. For example:

- For \(x = 0\), \(y = 0^2 - 1 = -1\)
- For \(x = 1\), \(y = 1^2 - 1 = 0\)
- For \(x = 2\), \(y = 2^2 - 1 = 3\)
- For \(x = 3\), \(y = 3^2 - 1 = 8\)
- For \(x = 4\), \(y = 4^2 - 1 = 15\)

This does not fit the data points either.

### Conclusion

The data points exhibit a linear relationship modeled perfectly by the equation \(y = -x - 1\). Hence, the best function to model the given data is:

\[ \text{linear; } y = -x - 1 \]