We are given a table of values for a linear function, and we need to determine the rule for the output when the input is $n$.

Table Summary:

• Input ($x$): -1, 2, 3, 4, $n$
• Output ($y$): 0, -2, -4, -6, ?

Step 1: Identify the pattern

The output is decreasing by 2 for every increase of 1 in the input. This indicates a linear relationship with a constant rate of change (slope).

• From $0$ to $-2$ (change of $-2$ when $x$ increases by 3).
• From $-2$ to $-4$ (change of $-2$).
• From $-4$ to $-6$ (change of $-2$).

This tells us that the slope (rate of change) is $-2$.

Step 2: Use the slope-intercept form

The general form of a linear function is:

$y = mx + b$

Where:

• $m$ is the slope (which we determined is $-2$),
• $b$ is the y-intercept.

To find $b$, we can use any known point from the table. Let's use the point $(-1, 0)$ where $x = -1$ and $y = 0$:

$0 = -2(-1) + b$ $0 = 2 + b$ $b = -2$

Step 3: Write the equation

The equation of the linear function is:

$y = -2x - 2$

Step 4: Substitute $n$ for $x$

When the input is $n$, the output is:

$y = -2n - 2$

Thus, the rule for the output when the input is $n$ is:

$y = -2n - 2$

Would you like any additional clarifications or details?

Here are 5 related questions to explore further:

1. How do you determine the slope of a linear function from a table?
2. What is the slope-intercept form of a linear equation?
3. How do you find the y-intercept of a linear function?
4. How can you use a linear equation to predict future values?
5. What are other forms of representing a linear function besides slope-intercept form?

Tip: To find the equation of a linear function from a table, always check for the constant rate of change (slope) and use a known point to find the y-intercept.