the linear function ​d(t)=7-6t 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 intercepts in this context.


Let's break this down step by step.

### Part (a): Sketch the Graph and Identify Domain and Range

The given linear function is \( d(t) = 7 - 6t \).

#### Graph:
1. This is a linear equation in slope-intercept form \( d(t) = mt + b \), where \( m = -6 \) (the slope) and \( b = 7 \) (the y-intercept).
2. To sketch the graph, note that the slope \( m = -6 \) indicates the line is decreasing, and the y-intercept \( b = 7 \) is the point where the graph intersects the y-axis.

   - **Intercept points**:
     - When \( t = 0 \): \( d(0) = 7 - 6(0) = 7 \), so the point is \( (0, 7) \).
     - Find the t-intercept (where \( d(t) = 0 \)): 
       \[
       7 - 6t = 0 \implies t = \frac{7}{6} \approx 1.17 \text{ hours}.
       \]
       So, the point is \( \left(\frac{7}{6}, 0\right) \).

3. The graph will be a straight line passing through these points: \( (0, 7) \) and \( \left(\frac{7}{6}, 0\right) \).

#### Domain and Range:
- **Domain**: In this context, \( t \) represents time after leaving the library. Since time can't be negative and likely only relevant until you reach home (i.e., when \( d(t) = 0 \)):
  \[
  0 \leq t \leq \frac{7}{6}
  \]
- **Range**: The distance \( d(t) \) decreases from 7 miles to 0 miles:
  \[
  0 \leq d(t) \leq 7
  \]

### Part (b): Intercepts and Their Meanings

- **Y-intercept**: \( (0, 7) \)
  - **Meaning**: This is the point where \( t = 0 \), meaning you are 7 miles away from home when you leave the library. The distance from home at the start is 7 miles.
  
- **X-intercept**: \( \left(\frac{7}{6}, 0\right) \)
  - **Meaning**: This is the point where \( d(t) = 0 \), meaning you have reached home after approximately 1.17 hours. The time taken to reach home from the library is 1.17 hours.

### Summary:
- The graph is a line decreasing from \( (0, 7) \) to \( \left(\frac{7}{6}, 0\right) \).
- Domain: \( 0 \leq t \leq \frac{7}{6} \)
- Range: \( 0 \leq d(t) \leq 7 \)
- The intercepts show the initial distance from home and the time taken to reach home.

Would you like to see a detailed sketch or have further questions?

### Five Related Questions:
1. How would the graph change if the slope were steeper, say \( -8t \) instead of \( -6t \)?
2. What happens to the intercepts if the y-intercept changes to 10 miles?
3. How does the domain and range change if the function represents the distance to a different location?
4. What would be the effect on the graph if the distance decreases at half the original rate?
5. Can this function be used to calculate when you are exactly halfway home?

### Tip:
Always remember that in real-world scenarios, the domain and range of a function are often constrained by the context, not just the math!

Explore how to graph the linear function d(t)=7-6t, understand its domain and range, and interpret intercepts in context. Suitable for high school students studying linear functions and graphing.

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