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The graph provided shows height \( d(t) \) in feet as a function of time \( t \) in seconds. It is a piecewise linear function, consisting of two distinct segments:

1. A line segment increasing from \( t = 0 \) to \( t = 6 \), reaching a peak at \( t = 6 \) with a height of 5 feet.
2. A line segment decreasing from \( t = 6 \) to \( t = 12 \), returning to a height of 0 feet.

The problem asks to find the rate of change over specific intervals:

### 17. Interval \( 3 \leq t \leq 6 \)
To find the rate of change (or slope), we use the formula for the slope of a line between two points \( (t_1, d(t_1)) \) and \( (t_2, d(t_2)) \):

\[
\text{Rate of change} = \frac{d(t_2) - d(t_1)}{t_2 - t_1}
\]

For \( 3 \leq t \leq 6 \), we know:
- At \( t = 3 \), the height is 2.5 feet (halfway up to 5).
- At \( t = 6 \), the height is 5 feet.

So, the rate of change is:

\[
\text{Rate of change} = \frac{5 - 2.5}{6 - 3} = \frac{2.5}{3} = 0.83 \, \text{feet per second}
\]

### 18. Interval \( 7 \leq t \leq 10 \)
For \( 7 \leq t \leq 10 \), we are on the decreasing part of the graph. Using the same method:

- At \( t = 7 \), the height is 4.17 feet (since it's decreasing from 5 feet at \( t = 6 \)).
- At \( t = 10 \), the height is 1.67 feet.

The rate of change is:

\[
\text{Rate of change} = \frac{1.67 - 4.17}{10 - 7} = \frac{-2.5}{3} = -0.83 \, \text{feet per second}
\]

So, the rate of change over this interval is negative, indicating a decrease in height.

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Would you like further details on these calculations or the slope interpretation?

Here are some follow-up questions to expand on the concepts:
1. How would the rate of change differ if the intervals included non-linear segments?
2. Can you calculate the rate of change over the entire interval \( 0 \leq t \leq 12 \)?
3. How does the slope relate to real-life interpretations, such as speed or velocity?
4. What is the significance of positive vs negative rates of change?
5. How can you find the rate of change for non-piecewise linear graphs?

**Tip:** In general, the rate of change of a linear function is constant, but for non-linear functions, the rate varies over different intervals.

For the graph shown, find the rate of change over the interval for 3 ≤ t ≤ 6 and 7 ≤ t ≤ 10.

Learn how to calculate the rate of change over the intervals 3 ≤ t ≤ 6 and 7 ≤ t ≤ 10 for a piecewise linear function representing height over time. This involves using the slope formula to determine the rate of increase and decrease over specific time intervals. Suitable for Grades 8-10 students learning about linear graphs and rates of change.

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