Let the function q be defined by `q(x)=a(x-h)^(2)`, where h is a positive constant, and a is a negative constant. For what value of x will the function q have its maximum value?

Options

A. `-h`
B. `-a`
C. a
D. h

Answer

To find the value of \( x \) for which the function \( q(x) = a(x - h)^2 \) has its maximum value, we can follow these steps:

### Step 1: Understand the function
The function \( q(x) = a(x - h)^2 \) is a quadratic function in the form of \( a(x - h)^2 \), where \( a \) is a negative constant and \( h \) is a positive constant.

### Step 2: Determine the nature of the quadratic function
Since \( a \) is negative, the parabola opens downwards. This means the function will have a maximum value at its vertex.

### Step 3: Identify the vertex of the parabola
The vertex of the function \( q(x) = a(x - h)^2 \) occurs at \( x = h \). This is because the expression \( (x - h)^2 \) reaches its minimum value (which is 0) when \( x = h \).

### Step 4: Calculate the maximum value of the function
Substituting \( x = h \) into the function:
\[ q(h) = a(h - h)^2 = a(0)^2 = 0 \]
Thus, the maximum value of \( q(x) \) is 0, which occurs at \( x = h \).

### Conclusion
The value of \( x \) for which the function \( q(x) \) has its maximum value is:
\[ \boxed{h} \]

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