Function a is defined by

a(x) =2-x2 - 16x +7.5, and function b si defined by b(x) = a(x - 7). What si the value of x when b(x) is at its maximum?

To find the value of \( x \) when \( b(x) \) is at its maximum, we need to analyze the function \( b(x) \) in terms of the given functions \( a(x) \) and \( b(x) = a(x - 7) \).

1. **Identify the vertex of \( a(x) \)**: The function \( a(x) = 2 - x^2 - 16x + 7.5 \) is a quadratic function of the form \( ax^2 + bx + c \). The vertex form of a quadratic function \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \).

   For the function \( a(x) = -x^2 - 16x + 9.5 \):
   - \( a = -1 \)
   - \( b = -16 \)

   The vertex \( x \) is:
   \[
   x = -\frac{b}{2a} = -\frac{-16}{2(-1)} = \frac{16}{-2} = -8
   \]

   Therefore, \( a(x) \) has its maximum value at \( x = -8 \).

2. **Transform to find \( b(x) \)**: The function \( b(x) = a(x - 7) \) represents a horizontal shift of the function \( a(x) \) by 7 units to the right. Thus, the vertex of \( b(x) \) will be at \( x = -8 + 7 \).

   Hence, the vertex (and maximum value) of \( b(x) \) occurs at:
   \[
   x = -8 + 7 = -1
   \]

Therefore, the value of \( x \) when \( b(x) \) is at its maximum is \( x = -1 \).

Would you like more details or have any other questions?

Here are 5 more questions you might find interesting:
1. How do you convert a quadratic function into its vertex form?
2. What is the significance of the vertex in a quadratic function?
3. How do you find the axis of symmetry for a quadratic function?
4. What are the different methods to solve quadratic equations?
5. How do transformations affect the graph of a function?

Tip: Understanding how horizontal and vertical shifts affect the graph of a function can help you quickly determine the vertex and other key points of transformed functions.

Discover how to find the maximum value of b(x) when it is derived from a transformed quadratic function. Learn step-by-step with detailed calculations and transformations. Suitable for high school students studying quadratic functions and transformations.

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