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Let's analyze the table and the required components for this exercise step by step:

### Given Table:
| \( t \) | \( p(t) \) | Ordered Pair |
|:------:|:---------:|:------------:|
| -2     | 8         | (-2, 8)      |
| -1     | 6         | (-1, 6)      |
| 0      | 4         | (0, 4)       |
| 1      | 2         | (1, 2)       |
| 2      | 0         | (2, 0)       |

### Step 1: Identify the Pattern
We can observe that for each step of increase in \( t \) (incrementing by 1), \( p(t) \) decreases by 2. This suggests that the function \( p(t) \) is linear.

### Step 2: Determine the Formula for \( p(t) \)
To find the symbolic rule for the function, we use the slope-intercept form of a line:
\[
p(t) = mt + b
\]
where \( m \) is the slope, and \( b \) is the y-intercept.

- **Slope (m)**: The slope \( m \) is the change in \( p(t) \) over the change in \( t \). Between points (-2, 8) and (-1, 6), the change in \( p(t) \) is -2, and the change in \( t \) is 1. Therefore, the slope is:
  \[
  m = \frac{-2}{1} = -2
  \]

- **Y-intercept (b)**: The y-intercept is the value of \( p(t) \) when \( t = 0 \). From the table, we see that when \( t = 0 \), \( p(t) = 4 \). Thus, \( b = 4 \).

So, the formula for \( p(t) \) is:
\[
p(t) = -2t + 4
\]

### Step 3: Graph the Function
- The points to plot are: (-2, 8), (-1, 6), (0, 4), (1, 2), and (2, 0).
- The graph will show a straight line with a slope of -2, passing through the y-axis at \( p(t) = 4 \).

### Step 4: Verbal Description
This function is **linear**. The relationship between the input (\( t \)) and output (\( p(t) \)) is that for every unit increase in \( t \), the value of \( p(t) \) decreases by 2.

### Summary:
- **Symbolic Rule**: \( p(t) = -2t + 4 \)
- **Verbal Description**: The function is linear, and as \( t \) increases by 1, \( p(t) \) decreases by 2.

Would you like me to explain how to plot these points on the graph?

Here are 5 related questions:
1. How can you verify that the function \( p(t) = -2t + 4 \) fits all the points in the table?
2. What happens to the value of \( p(t) \) if \( t \) increases beyond 2?
3. How would the graph change if the slope were positive instead of negative?
4. What would the function look like if the y-intercept were 0?
5. How does the slope of -2 affect the steepness of the line?

**Tip**: Always check if your graph passes through all plotted points to ensure accuracy when graphing a linear function.

Identify the pattern from the table, use that information to construct the graph and determine the formula for the function p(t). Then use words to describe the relationship between the input and output variables.

Learn how to identify linear patterns from a table of values and construct a graph. Solve for the function p(t) in slope-intercept form, determine the rule for the function, and understand the relationship between input t and output p(t). Suitable for Grades 7-9 students.

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