https://file.aiquickdraw.com/mathbot/file/1726775452571yzy9gyzt.jpg

Let's analyze the table of values in the image.

We are given the following inputs and outputs for a linear function:

| Input  | Output |
|--------|--------|
| 1      | -4     |
| 2      | 0      |
| 3      | 4      |
| 4      | 8      |
| n      | ?      |

To find the pattern, notice the output changes with a constant difference as the input increases by 1. Specifically:

- From input 1 to input 2: \( 0 - (-4) = 4 \)
- From input 2 to input 3: \( 4 - 0 = 4 \)
- From input 3 to input 4: \( 8 - 4 = 4 \)

The output increases by 4 for every increase of 1 in the input, confirming this is a linear function with a constant rate of change.

### Step 1: Determine the slope \( m \)
The slope (rate of change) is 4, which means the linear function has the form:

\[
y = 4x + b
\]

### Step 2: Find the y-intercept \( b \)
Using the point (1, -4):

\[
-4 = 4(1) + b \implies -4 = 4 + b \implies b = -8
\]

Thus, the equation of the linear function is:

\[
y = 4x - 8
\]

### Step 3: Calculate the output for input \( n \)
The output when the input is \( n \) is:

\[
y = 4n - 8
\]

Thus, the output for the input \( n \) is \( 4n - 8 \).

Would you like more details or clarifications? Here are some related questions to consider:

1. How can you use two points to find the equation of a line?
2. Why is the slope the same between any two consecutive points on a linear function?
3. How do we determine the y-intercept from a table of values?
4. Can this method be used for other types of functions (e.g., quadratic, exponential)?
5. How do you apply this function to predict other outputs beyond those in the table?

**Tip:** Always check for a constant rate of change when analyzing tables of linear functions. It ensures the function is indeed linear.

A table of values of a linear function is shown below. Find the output when the input is n.

Learn how to find the output for a given input 'n' using the table of values for a linear function. Explore key algebraic concepts like slope and y-intercept to form the equation y = 4x - 8. Suitable for students in Grades 7-9.

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