Let's analyze the table in the new image:

$x$	1	2	3	4
$y$	4	1	-2	-5

To find the function rule, let's first examine if there is a pattern. We can observe that the values of $y$ are decreasing in a regular pattern. Specifically:

• From $4$ to $1$ (decrease by 3),
• From $1$ to $-2$ (decrease by 3),
• From $-2$ to $-5$ (decrease by 3).

This constant rate of change suggests that the function is linear, in the form $y = mx + b$, where:

• $m$ is the slope (rate of change), and
• $b$ is the y-intercept (value of $y$ when $x = 0$).

Step 1: Find the slope $m$

The slope $m$ is the change in $y$ divided by the change in $x$: $m = \frac{\Delta y}{\Delta x} = \frac{-5 - (-2)}{4 - 3} = \frac{-3}{1} = -3$

So, the slope is $-3$.

Step 2: Use a point to find the y-intercept $b$

Now that we have the slope, we can use one of the points from the table to solve for $b$. Let's use the point $(1, 4)$.

Using the equation $y = mx + b$, substitute $x = 1$, $y = 4$, and $m = -3$:

$4 = -3(1) + b \quad \Rightarrow \quad 4 = -3 + b \quad \Rightarrow \quad b = 7$

Final Function Rule

The function rule is: $y = -3x + 7$

Would you like further details or have any questions?

Here are 5 related questions:

1. How can you graph the function $y = -3x + 7$ based on the table?
2. What happens to the slope if the differences in $y$ values aren't constant?
3. How can you determine the y-intercept if the table doesn’t provide $x = 0$?
4. How does the slope $m = -3$ affect the steepness and direction of the graph?
5. How would the function rule change if the table had quadratic data?

Tip: For linear functions, always check if the differences between consecutive $y$-values are constant to confirm a linear relationship.