3. temperature throughout the day the temperature ( t(h) ) during a spring day can be modeled by ( t(h) = -(h - 12)^2 + 75 ), where ( t ) is the temperature in fahrenheit and ( h ) is the hour (using 24-hour time).
a. what parent function is this based on?
b. explain the meaning of each transformation
c. what is the maximum temperature and when does it occur?
d. how does this compare to ( t(h) = -h^2 + 75 )?
e. what would the new function be if the peak temperature was ( 80^circ )f?

Question

3. temperature throughout the day the temperature ( t(h) ) during a spring day can be modeled by ( t(h) = -(h - 12)^2 + 75 ), where ( t ) is the temperature in fahrenheit and ( h ) is the hour (using 24-hour time).
   a. what parent function is this based on?
   b. explain the meaning of each transformation
   c. what is the maximum temperature and when does it occur?
   d. how does this compare to ( t(h) = -h^2 + 75 )?
   e. what would the new function be if the peak temperature was ( 80^circ )f?

Accepted Answer

### Part (a)
# Explanation:
## Step1: Identify Parent Function
The given function is $ T(h) = -(h - 12)^2 + 75 $. The general form of a quadratic function is $ y = ax^2+bx + c $, and the simplest quadratic function (parent function) is $ y = x^2 $. The given function is a transformed version of $ y = x^2 $ (since it has a square of a linear term in $ h $, with transformations like horizontal shift, vertical shift, and reflection).
# Answer:
The parent function is the quadratic function $ y = x^2 $ (or $ f(x)=x^2 $).

### Part (b)
# Explanation:
## Step1: Analyze Coefficient of Squared Term
The coefficient of $ (h - 12)^2 $ is $ - 1 $. A negative coefficient indicates a reflection over the horizontal axis (the $ h $-axis, or in terms of the function's graph, it opens downward instead of upward like the parent function $ y = x^2 $).
## Step2: Analyze Horizontal Shift
The term $ (h - 12) $ inside the square means there is a horizontal shift. For a function $ y=(x - k)^2 $, the graph shifts $ k $ units to the right. Here, $ k = 12 $, so the graph of the parent function $ y = x^2 $ is shifted 12 units to the right (along the $ h $-axis, which represents hours).
## Step3: Analyze Vertical Shift
The $ + 75 $ at the end of the function means there is a vertical shift. For a function $ y = f(x)+k $, the graph shifts $ k $ units up. Here, $ k = 75 $, so the graph is shifted 75 units up (along the $ T $-axis, which represents temperature in Fahrenheit).
# Answer:
- Reflection: The negative sign in front of $ (h - 12)^2 $ reflects the graph of the parent function $ y = x^2 $ over the horizontal axis (opens downward).
- Horizontal Shift: The $ (h - 12) $ shifts the graph 12 units to the right (along the hour axis).
- Vertical Shift: The $ + 75 $ shifts the graph 75 units up (along the temperature axis).

### Part (c)
# Explanation:
## Step1: Recall Properties of Quadratic Function
The function $ T(h)=-(h - 12)^2 + 75 $ is a quadratic function in vertex form $ y=a(x - h)^2 + k $, where $ (h,k) $ is the vertex. Since $ a=-1<0 $, the parabola opens downward, so the vertex is the maximum point.
## Step2: Identify Vertex Coordinates
In the function $ T(h)=-(h - 12)^2 + 75 $, comparing with $ y=a(x - h)^2 + k $, we have $ h = 12 $ (the $ h $-coordinate of the vertex, representing the hour) and $ k = 75 $ (the $ T $-coordinate of the vertex, representing the temperature).
# Answer:
The maximum temperature is $ 75^{\circ}\text{F} $, and it occurs at $ h = 12 $ (which is 12:00 in 24 - hour time, or noon).

### Part (d)
# Explanation:
## Step1: Analyze $ T(h)=-h^2 + 75 $
The function $ T(h)=-h^2 + 75 $ is also a quadratic function in vertex form with $ a=-1 $, $ h = 0 $ (since it can be written as $ T(h)=-(h - 0)^2 + 75 $) and $ k = 75 $.
## Step2: Compare with $ T(h)=-(h - 12)^2 + 75 $
- **Reflection and Vertical Shift**: Both functions have $ a=-1 $ (so same reflection over the horizontal axis) and $ k = 75 $ (same vertical shift up by 75 units).
- **Horizontal Shift**: The function $ T(h)=-h^2 + 75 $ has its vertex at $ h = 0 $ (so the maximum temperature occurs at $ h = 0 $, or midnight), while $ T(h)=-(h - 12)^2 + 75 $ has its vertex at $ h = 12 $ (maximum at noon). So the main difference is the horizontal shift: $ T(h)=-(h - 12)^2 + 75 $ is the graph of $ T(h)=-h^2 + 75 $ shifted 12 units to the right.
# Answer:
Both functions are downward - opening parabolas with the same vertical shift (up 75 units) and same reflection (over the horizontal axis). The function $ T(h)=-(h - 12)^2 + 75 $ is the function $ T(h)=-h^2 + 75 $ shifted 12 units to the right. The maximum temperature for both is $ 75^{\circ}\text{F} $, but it occurs at $ h = 12 $ (noon) for $ T(h)=-(h - 12)^2 + 75 $ and at $ h = 0 $ (midnight) for $ T(h)=-h^2 + 75 $.

### Part (e)
# Explanation:
## Step1: Recall Vertex Form of Quadratic Function
The original function is in vertex form $ T(h)=a(h - h_0)^2 + k $, where $ (h_0,k) $ is the vertex. The peak temperature (maximum temperature) occurs at the vertex, so $ k $ is the peak temperature.
## Step2: Determine New Function
In the original function $ T(h)=-(h - 12)^2 + 75 $, the peak temperature is $ 75^{\circ}\text{F} $ (the $ k $-value). If the peak temperature is $ 80^{\circ}\text{F} $, we need to change the $ k $-value from 75 to 80 while keeping the other transformations (reflection, horizontal shift) the same. So the new function will be $ T(h)=-(h - 12)^2 + 80 $.
# Answer:
The new function is $ T(h)=-(h - 12)^2 + 80 $.

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Identify Parent Function

Step1: Analyze Coefficient of Squared Term

Step2: Analyze Horizontal Shift

Step3: Analyze Vertical Shift

Step1: Recall Properties of Quadratic Function

Step2: Identify Vertex Coordinates

Answer:

Part (b)