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.