Question

Write a function rule that models this situation x values: 1, 3, 6, 7, y values: 21, 15, 6, 3

Answer 1

To model the given values, we can look for a function rule that relates the x values to the corresponding y values. Let's first tabulate the data you provided:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 21 \\
3 & 15 \\
6 & 6 \\
7 & 3 \\
\hline
\end{array}
\]

Next, we will try to find a pattern or a mathematical function connecting the x and y values. Given the values, it looks like the y values decrease as x increases.

We can analyze the differences between the y values for corresponding x values:

1. From \(1\) to \(3\) (increase of 2 in \(x\)): \(21\) to \(15\) (decrease of 6)
2. From \(3\) to \(6\) (increase of 3 in \(x\)): \(15\) to \(6\) (decrease of 9)
3. From \(6\) to \(7\) (increase of 1 in \(x\)): \(6\) to \(3\) (decrease of 3)

We can also look for a polynomial function that might work. One approach is to use polynomial regression, but let's see if we can find a simple rule manually.

Considering that we have 4 data points, we can try to formulate a piecewise function, or find a polynomial that fits the data.

Let's try fitting a linear function of the form \(y = ax + b\). Using the first and last points (1, 21), (7, 3) to create a rough estimate:

Using the two points:
\[
(1, 21) \text{ and } (7, 3)
\]

The slope \(m\) can be computed as:
\[
m = \frac{3 - 21}{7 - 1} = \frac{-18}{6} = -3
\]

Using the point-slope form:
\[
y - 21 = -3(x - 1)
\]

This simplifies to:
\[
y = -3x + 24
\]

Now we will check if this fits all points:

- For \(x = 1: y = -3(1) + 24 = 21\) (correct)
- For \(x = 3: y = -3(3) + 24 = 15\) (correct)
- For \(x = 6: y = -3(6) + 24 = 6\) (correct)
- For \(x = 7: y = -3(7) + 24 = 3\) (correct)

Thus, the function rule that models the situation is:

\[
\boxed{y = -3x + 24}
\]