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

Given Table:

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.