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The problem involves a linear function \( d(t) = 9 - 8t \), where \( d \) represents the distance from home (in miles), and \( t \) is the time (in hours) after you leave the library.

Let's tackle each part:

### Part (a): Sketch the graph of the function

The equation \( d(t) = 9 - 8t \) is a linear function with:
- **Slope**: \( -8 \) (which means for every hour \( t \), the distance decreases by 8 miles).
- **y-intercept**: \( d(0) = 9 \), meaning you are 9 miles from home at \( t = 0 \).

#### Domain and Range:
- **Domain**: The context involves real-world time after leaving the library, so \( t \geq 0 \). Hence, the domain is \( t \in [0, \infty) \).
- **Range**: Since the distance cannot be negative and decreases with time, the range is \( d(t) \in [0, 9] \).

#### Sketch:
I'll describe how to plot this on the graph:
1. At \( t = 0 \), \( d(0) = 9 \), so the point (0, 9) is on the graph.
2. At \( t = 1 \), \( d(1) = 9 - 8(1) = 1 \), so the point (1, 1) is on the graph.
3. At \( t = 9/8 \approx 1.125 \), the function will reach zero.

So, the line starts at (0, 9) and slopes downward with a rate of -8. You can stop the graph once the distance reaches zero because you cannot go beyond zero miles.

### Part (b): Intercepts of the graph
- **y-intercept**: This is the point where \( t = 0 \). From the equation, \( d(0) = 9 \), so the y-intercept is \( (0, 9) \). This means you start 9 miles from home when leaving the library.
- **x-intercept**: Set \( d(t) = 0 \) to find when the distance becomes zero. Solving \( 9 - 8t = 0 \) gives \( t = 9/8 \approx 1.125 \). This is the time when you arrive home (approximately 1.125 hours after leaving the library).

### Explanation of intercepts:
- The y-intercept of 9 represents the initial distance from home (9 miles) when you leave the library.
- The x-intercept of 1.125 hours represents the time it takes to reach home, after traveling 9 miles.

Would you like further details or any clarifications?

### Relative questions:
1. What would the graph look like if the slope were positive instead of negative?
2. How would the domain and range change if the function represented a different context (e.g., starting distance increases)?
3. How can we interpret the slope in practical terms for different scenarios?
4. What happens if the y-intercept changes, but the slope remains the same?
5. How would you solve for \( t \) if the function was quadratic rather than linear?

### Tip:
When graphing a linear function, always identify the slope and intercepts first; they provide crucial points for accurate plotting.

The linear function d(t) = 9 - 8t gives your distance d (in miles) from home, t hours after you leave the library. a. Sketch a graph of the function. Write the domain and range in inequalities for d(t) in this context. b. What are the intercepts of the graph? Explain the meaning of any.

Learn how to graph the linear function d(t) = 9 - 8t, determine the domain and range, and find the x and y intercepts. This problem is designed for students in Grades 9-11 and focuses on key concepts in graphing linear functions.

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