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I see the image contains a math problem that is based on HVAC systems and temperature fluctuations in Dallas. The problem is asking you to work with different mathematical functions, particularly in terms of air blowing through vents and energy usage, with questions involving the definition and manipulation of piecewise functions, sketching graphs, and energy calculations.

Let me walk you through the steps of solving the problem.

### Part (a): Sketch \( A_1(t) \)
1. **Function given**: \( A_1(t) = 5\left( \frac{1}{2}t - 1 \right)^2 \)
2. **Find the domain**: The domain is between 12 am (midnight) and 4 am, or \( 0 \leq t < 4 \).
3. **Find the range**: Since this is a quadratic function, it opens upwards. The minimum value occurs at the vertex, which can be found by setting the derivative equal to zero or by identifying the minimum in the quadratic form.

### Part (b): Sketch \( A_2(t) \)
1. **Function given**: \( A_2(t) = \frac{1}{4} \)
2. **Domain**: This is a constant function from 4 am to 8 am, or \( 4 \leq t < 8 \).
3. **Range**: Since this is a constant function, the range is simply \( \frac{1}{4} \).

### Part (c): Sketch \( A_3(t) \)
1. **Function given**: \( A_3(t) = 2\sqrt{t - 8} + \frac{1}{4} \)
2. **Domain**: The domain is from 8 am to 12 am the next day, or \( 8 \leq t \leq 24 \) (midnight is excluded).
3. **Range**: Since the square root function grows, the range of this function will depend on the minimum and maximum values within the domain.

### Part (d): Piecewise Function \( A(t) \)
You are asked to write the complete function \( A(t) \) as a piecewise function, combining the three separate pieces \( A_1(t), A_2(t), A_3(t) \), each defined over a specific domain:
\[
A(t) = \begin{cases} 
A_1(t), & 0 \leq t < 4 \\
A_2(t), & 4 \leq t < 8 \\
A_3(t), & 8 \leq t < 24 
\end{cases}
\]
Ensure you clearly write the inequalities that describe the domain for each part.

### Part (e): Sketching \( A(t) \)
You need to sketch the piecewise function \( A(t) \), showing each of the three sections described above. Focus on the transitions between the sections, ensuring that they match at the boundary points \( t = 4 \) and \( t = 8 \).

### Part (f): Explaining Why \( A(t) \) is a Function
Explain that \( A(t) \) assigns exactly one value of air blowing for each hour \( t \), satisfying the definition of a function. You'll need to calculate specific values like \( A(4) \) and \( A(8) \) to show how the function behaves at these critical points.

### Part (g): Changing the Domain
This part involves brainstorming changes in the domain that would cause \( A(t) \) to no longer be a function. For example, if two values of air blowing were assigned to the same time \( t \), it would violate the function definition.

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Let me know if you'd like detailed solutions or explanations for any of these parts, and feel free to ask questions!

Here are 5 related questions that extend from this:

1. What are the physical implications of changing the piecewise function at different intervals?
2. How does the rate of air blowing through vents affect the building's energy usage?
3. What would happen if \( A(t) \) were discontinuous at one of the boundary points (e.g., \( t = 4 \))?
4. How would the range of each function change if the total air volume had to be reduced by half?
5. What would be the effect of increasing the value of the constant in \( A_2(t) \) on the total air output?

**Tip**: When sketching piecewise functions, always check for continuity at the points where the pieces join. This ensures a smooth transition across intervals.

Determine the amount of energy that HVAC units will use to keep the building interior at 74°F in Dallas, given temperature fluctuations and air blowing functions over time. Work with functions to describe air blowing rates, define a piecewise function, sketch graphs, and calculate energy usage.

Explore how to calculate the energy usage of HVAC units based on temperature fluctuations in Dallas. Learn to define piecewise functions for air blowing rates, sketch graphs, analyze function domains and ranges, and calculate energy consumption. Suitable for high school students (Grades 10-12).

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