Question

a quadratic function $f(x)$ is hidden from view. you must find the maximum value of the function $f(x)$. choose the form of the quadratic function $f(x)$ that you would like to see in order to answer the question most efficiently. form: select a form

Explanation:

To determine the maximum value of a quadratic function \( f(x) \), the most efficient form to use is the vertex form (also known as the standard form for quadratic functions in vertex form), which is given by:

$$ f(x) = a(x - h)^2 + k $$

• In the vertex form \( f(x) = a(x - h)^2 + k \):
• The vertex of the parabola is at the point \( (h, k) \).
• The coefficient \( a \) determines whether the parabola opens upward or downward:
• If \( a > 0 \), the parabola opens upward, and the vertex \( (h, k) \) is the minimum point.
• If \( a < 0 \), the parabola opens downward, and the vertex \( (h, k) \) is the maximum point.
• Therefore, by examining the vertex form:
• If \( a < 0 \), the maximum value of the function is \( k \) (the \( y \)-coordinate of the vertex).
• If \( a > 0 \), the function has a minimum value (not a maximum), but the vertex form still allows us to identify this immediately.

To find the maximum value of \( f(x) \) efficiently, we should choose the vertex form \( \boldsymbol{f(x) = a(x - h)^2 + k} \). This form directly reveals the vertex \( (h, k) \), and if \( a < 0 \), the maximum value is \( k \).

Answer:

• In the vertex form \( f(x) = a(x - h)^2 + k \):
• The vertex of the parabola is at the point \( (h, k) \).
• The coefficient \( a \) determines whether the parabola opens upward or downward:
• If \( a > 0 \), the parabola opens upward, and the vertex \( (h, k) \) is the minimum point.
• If \( a < 0 \), the parabola opens downward, and the vertex \( (h, k) \) is the maximum point.
• Therefore, by examining the vertex form:
• If \( a < 0 \), the maximum value of the function is \( k \) (the \( y \)-coordinate of the vertex).
• If \( a > 0 \), the function has a minimum value (not a maximum), but the vertex form still allows us to identify this immediately.

To find the maximum value of \( f(x) \) efficiently, we should choose the vertex form \( \boldsymbol{f(x) = a(x - h)^2 + k} \). This form directly reveals the vertex \( (h, k) \), and if \( a < 0 \), the maximum value is \( k \).