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1 answer

To analyze the equation \( A(w) = w^2 + 5w \) and determine the relationship between \( A \) and \( w \), we can review the behavior of the function.

  1. Identify the nature of the function: \( A(w) \) is a quadratic function of the form \( A(w) = w^2 + 5w \).

  2. Determine the shape of the graph: Since the coefficient of \( w^2 \) is positive, the graph opens upwards, which usually implies it has a minimum value rather than a maximum.

  3. Finding the vertex: The vertex of a quadratic function in the form \( A(w) = aw^2 + bw + c \) can be found at the point \( w = -\frac{b}{2a} \):

    • Here, \( a = 1 \) and \( b = 5 \).
    • Therefore, \( w = -\frac{5}{2(1)} = -\frac{5}{2} = -2.5 \).

  4. Analyze the function behavior:

    • As \( w \) increases from \( -\infty \) to the vertex \( w = -2.5 \), \( A(w) \) decreases.
    • As \( w \) increases from \( -2.5 \) to \( +\infty \), \( A(w) \) increases.

  5. Select appropriate responses:

    • \( A(w) \) does not have a maximum value in the positive \( w \) domain and indeed it decreases for negative \( w \) and increases for \( w > -2.5 \).
    • Therefore, the accurate statement regarding the behavior, when \( w > 0 \), is:
    • \( A(w) \) increases as \( w \) increases when \( w > 0. \)

The best answer from the given options is:

\( A(w) \) increases as \( w \) increases when \( w > 0. \)