Questions: Identify the vertex and sketch the graph. f(x)=-2 x^2-24 x-72

Identify the vertex and sketch the graph.
f(x)=-2 x^2-24 x-72
Transcript text: Identify the vertex and sketch the graph. \[ f(x)=-2 x^{2}-24 x-72 \]
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Solution

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Solution Steps

Step 1: Identify the Vertex of the Parabola

The given quadratic function is \( f(x) = -2x^2 - 24x - 72 \). The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. To find the vertex, we use the formula for the x-coordinate of the vertex, \( h = -\frac{b}{2a} \).

Here, \( a = -2 \) and \( b = -24 \).

\[ h = -\frac{-24}{2 \times -2} = -\frac{24}{-4} = 6 \]

Now, substitute \( x = 6 \) back into the function to find \( k \).

\[ f(6) = -2(6)^2 - 24(6) - 72 = -2(36) - 144 - 72 = -72 - 144 - 72 = -288 \]

Thus, the vertex is \((6, -288)\).

Step 2: Write the Function in Vertex Form

The vertex form of the function is:

\[ f(x) = -2(x - 6)^2 - 288 \]

Final Answer

The vertex of the function \( f(x) = -2x^2 - 24x - 72 \) is \((6, -288)\).

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