Question

what is the \\(y\\)-value of the vertex of the function \\(f(x)=-(x-3)(x+11)\\)?

\\(\bigcirc\\) \\(-8\\)
\\(\bigcirc\\) \\(-4\\)
\\(\bigcirc\\) \\(33\\)
\\(\bigcirc\\) \\(49\\)

Explanation:

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<reasoning>

Find the x-intercepts of the quadratic function

\[

$$\begin{aligned} &f(x) = -(x-3)(x+11) \\ &x_1 = 3, \quad x_2 = -11 \end{aligned}$$

\]

Calculate the x-coordinate of the vertex

\[

$$\begin{aligned} x_v &= \frac{x_1 + x_2}{2} \\ x_v &= \frac{3 + (-11)}{2} = \frac{-8}{2} = -4 \end{aligned}$$

\]

Evaluate the function at the vertex x-coordinate to find the y-value

\[

$$\begin{aligned} y_v &= f(-4) \\ y_v &= -(-4 - 3)(-4 + 11) \\ y_v &= -(-7)(7) \\ y_v &= 49 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>-8</mcq-option>
<mcq-option>-4</mcq-option>
<mcq-option>33</mcq-option>
<mcq-correct>49</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Vertex"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Find the x-intercepts of the quadratic function

\[

$$\begin{aligned} &f(x) = -(x-3)(x+11) \\ &x_1 = 3, \quad x_2 = -11 \end{aligned}$$

\]

Calculate the x-coordinate of the vertex

\[

$$\begin{aligned} x_v &= \frac{x_1 + x_2}{2} \\ x_v &= \frac{3 + (-11)}{2} = \frac{-8}{2} = -4 \end{aligned}$$

\]

Evaluate the function at the vertex x-coordinate to find the y-value

\[

$$\begin{aligned} y_v &= f(-4) \\ y_v &= -(-4 - 3)(-4 + 11) \\ y_v &= -(-7)(7) \\ y_v &= 49 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>-8</mcq-option>
<mcq-option>-4</mcq-option>
<mcq-option>33</mcq-option>
<mcq-correct>49</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
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"Quadratic Vertex"
]
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