Question

A graph has the points (4, -5), (8, -1), (9, 0), and (10, -1). What is the function rule?

Answer 1

To find a function rule that fits the given points \((4, -5)\), \((8, -1)\), \((9, 0)\), and \((10, -1)\), we can start by observing the points to see if they might fit a polynomial function, particularly a quadratic function, as we have four points.

A quadratic function can be expressed in the form:

\[
f(x) = ax^2 + bx + c
\]

where \(a\), \(b\), and \(c\) are constants.

Given that we have four points, we can set up a system of equations based on these points to solve for \(a\), \(b\), and \(c\).

1. For the point \((4, -5)\):
\[
16a + 4b + c = -5 \quad \text{(1)}
\]

2. For the point \((8, -1)\):
\[
64a + 8b + c = -1 \quad \text{(2)}
\]

3. For the point \((9, 0)\):
\[
81a + 9b + c = 0 \quad \text{(3)}
\]

4. For the point \((10, -1)\):
\[
100a + 10b + c = -1 \quad \text{(4)}
\]

We can solve this system of equations. We will first use equations (1), (2), and (3) to find \(a\), \(b\), and \(c\).

Subtract equation (1) from equation (2):
\[
(64a + 8b + c) - (16a + 4b + c) = -1 + 5
\]
\[
48a + 4b = 4 \quad \Rightarrow \quad 12a + b = 1 \quad \text{(5)}
\]

Subtract equation (2) from equation (3):
\[
(81a + 9b + c) - (64a + 8b + c) = 0 + 1
\]
\[
17a + b = 1 \quad \text{(6)}
\]

Now we have two new equations (5) and (6):
1. \(12a + b = 1\)
2. \(17a + b = 1\)

Subtract equation (5) from equation (6):
\[
(17a + b) - (12a + b) = 1 - 1
\]
\[
5a = 0 \quad \Rightarrow \quad a = 0
\]

Now substitute \(a = 0\) back into equation (5):
\[
12(0) + b = 1 \quad \Rightarrow \quad b = 1
\]

Now substitute \(a = 0\) and \(b = 1\) back into equation (1):
\[
16(0) + 4(1) + c = -5 \quad \Rightarrow \quad 4 + c = -5 \quad \Rightarrow \quad c = -9
\]

Thus, the function rule is:
\[
f(x) = 0x^2 + 1x - 9 \quad \Rightarrow \quad f(x) = x - 9
\]

Let's verify if the function \(f(x) = x - 9\) fits all given points:
- For \(x = 4\): \(f(4) = 4 - 9 = -5\) ✔
- For \(x = 8\): \(f(8) = 8 - 9 = -1\) ✔
- For \(x = 9\): \(f(9) = 9 - 9 = 0\) ✔
- For \(x = 10\): \(f(10) = 10 - 9 = 1\) ✔

The point \((10, -1)\) does not fit, suggesting that either the function might not be completely indicative of the pattern (which does also indicate that the fourth point likely follows a different rule or measures from a different context).

The derived function \(f(x) = x - 9\) gives correct outputs for the first three points but fails to encapsulate the last point as initially posited. Therefore, while \(f(x) = x - 9\) represents a viable function rule for a substantial subset of the points, further investigation would be needed to accurately seed the fourth point's function dependency or context.

To finalize, a polynomial attempt results in the conclusion:
The simplest function rule compliant with three of four points is:
\[
f(x) = x - 9
\]
with context-aware adjustments potentially necessary considering outlier behavior in fourth point.