Question

1. 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) = -h² + 75?
e. what would the new function be if the peak temperature was 80°f?

Explanation:

Part (a)

Step1: Identify Parent Function

The given function is \( T(h)=-(h - 12)^2+75 \). The general form of a quadratic function (parabola) 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 term with transformations like horizontal shift, vertical shift, and reflection).

• Reflection: The negative sign in front of \( (h - 12)^2 \) (i.e., \( a=-1 \)) reflects the parent function \( y = x^2 \) over the horizontal axis (x - axis). This makes the parabola open downward, meaning the temperature function has a maximum value (since it opens down).
• Horizontal Shift: The \( (h - 12) \) term indicates a horizontal shift. For a function \( y=(x - k)^2 \), it shifts the graph of \( y = x^2 \) \( k \) units to the right. Here, \( k = 12 \), so the graph of the parent function \( y = x^2 \) is shifted 12 units to the right. In the context of temperature, this means the "peak" of the temperature (the vertex of the parabola) occurs at \( h = 12 \) (12:00 in 24 - hour time).
• Vertical Shift: The \( + 75 \) term at the end is a vertical shift. For a function \( y=x^2 + c \), it shifts the graph of \( y = x^2 \) \( c \) units up. Here, \( c = 75 \), so the graph of the transformed function is shifted 75 units up. In terms of temperature, this means the base (vertex) of the parabola (before considering the horizontal shift) is at \( y = 75 \) (the temperature at the vertex, before horizontal shift, is 75°F, but with the horizontal shift, the vertex is at \( h = 12 \), \( T = 75 \) initially, but we will see in part (c) that the vertex is the maximum).

Step1: Recall Vertex Form of Quadratic

The vertex form of a quadratic function is \( y=a(x - h)^2+k \), where \( (h,k) \) is the vertex of the parabola. If \( a<0 \), the parabola opens downward, and the vertex is the maximum point.

Step2: Identify Vertex from Given Function

For the function \( T(h)=-(h - 12)^2+75 \), comparing with \( y=a(x - h)^2+k \), we have \( a=-1 \), \( h = 12 \), and \( k = 75 \). Since \( a=-1<0 \), the vertex \( (12,75) \) is the maximum point.

Answer:

The parent function is the quadratic function \( y = x^2 \) (or \( f(x)=x^2 \)).

Part (b)