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Let's break down each part of the assignment step by step:

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**1. The following graph models the braking distance, \( B \) (in feet), as a function of the speed, \( s \), of a vehicle (in miles per hour). Use the graph to answer the questions that follow.**

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### a. Explain why the graph represents a function.
- A graph represents a function if each input value (in this case, speed \( s \)) corresponds to exactly one output value (in this case, braking distance \( B(s) \)). In other words, for every value of \( s \), there should be one unique \( B(s) \). The graph in question appears to pass the vertical line test, meaning that no vertical line intersects the graph at more than one point, thus confirming it represents a function.

### b. Express the input/output using function notation. The function name is \( p \).
- The braking distance \( B \) as a function of speed \( s \) can be expressed as \( p(s) = B(s) \). Here, \( p \) is the function name and \( s \) is the input (speed), while \( B(s) \) is the braking distance, which is the output.

### c. Use the graph to solve \( B(s) = 50 \) and explain the meaning using the problem context.
- To solve \( B(s) = 50 \), find the point on the graph where the braking distance \( B(s) \) is equal to 50 feet, and then find the corresponding value of \( s \) (speed). From the graph, it seems like \( B(s) = 50 \) happens when the speed \( s \) is approximately 35 miles per hour. This means that when a vehicle is traveling at 35 mph, the braking distance required is 50 feet.

### d. Use the graph to evaluate \( B(20) \) and explain the meaning using the problem context.
- From the graph, find the value of \( B(s) \) when \( s = 20 \). It appears that \( B(20) \) is approximately 16.8 feet. This means that when the vehicle is traveling at 20 mph, the braking distance required is about 16.8 feet.

### e. The function that this graph is modeling is \( B(s) = 0.042s^2 \). Use the intersect method in Desmos, solve \( B(s) = 100 \) and state your answer to 3 decimal places. State the meaning of the solution.
- The equation is \( B(s) = 0.042s^2 \), and you need to solve \( B(s) = 100 \).
  
  \[
  0.042s^2 = 100
  \]
  To solve for \( s \):
  \[
  s^2 = \frac{100}{0.042} \approx 2380.95
  \]
  \[
  s = \sqrt{2380.95} \approx 48.81
  \]
  So, \( s \approx 48.808 \) miles per hour (rounded to three decimal places).

- **Meaning**: The solution \( s = 48.808 \) means that when a vehicle is traveling at approximately 48.8 mph, the required braking distance is 100 feet.

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Would you like more details on any part of this, or have any specific questions?

### Follow-up Questions:
1. Can you explain how to perform the vertical line test in more detail?
2. How can you use a graphing calculator like Desmos to find the intersection point?
3. What does it mean when a function fails the vertical line test?
4. How does the shape of the graph (quadratic) relate to the physical concept of braking distance?
5. Can you explain why \( s = 48.808 \) makes sense for the braking distance context?

### Tip:
Always double-check the scale of the graph to ensure accurate reading of points when solving equations graphically.

The following graph models the braking distance, B (in feet), as a function of the speed, s, of a vehicle (in miles per hour). Use the graph to answer the questions that follow.

Learn how to solve a quadratic equation modeling braking distance as a function of vehicle speed. Understand key concepts such as function representation, graph interpretation, and solving quadratic equations. Suitable for high school students in Grades 9-12.

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