Question

find the vertex of this quadratic function.

\\y = -3x^2 - 30x - 85\\

(?, )

Explanation:

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<reasoning>

Identify coefficients from standard form

Using the Quadratic Standard Form knowledge point
\[

$$\begin{aligned} &y = ax^2 + bx + c\\ &y = -3x^2 - 30x - 85\\ &a = -3,\quad b = -30,\quad c = -85 \end{aligned}$$

\]

Calculate the x-coordinate of the vertex

Using the Axis of Symmetry knowledge point
\[

$$\begin{aligned} &x = -\frac{b}{2a}\\ &x = -\frac{-30}{2(-3)} = -\frac{-30}{-6} = -5 \end{aligned}$$

\]

Calculate the y-coordinate of the vertex

Substitute \(x = -5\) into the original quadratic equation:
\[

$$\begin{aligned} y &= -3(-5)^2 - 30(-5) - 85\\ y &= -3(25) + 150 - 85\\ y &= -75 + 150 - 85\\ y &= 75 - 85\\ y &= -10 \end{aligned}$$

\]

State the vertex coordinates

The coordinates of the vertex \((h, k)\) are:
\[
(h, k) = (-5, -10)
\]
</reasoning>

<answer>
Find the vertex of this quadratic function.
\(y = -3x^2 - 30x - 85\)
(<blank>-5</blank>, <blank>-10</blank>)
</answer>

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Answer:

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<reasoning>

Identify coefficients from standard form

Using the Quadratic Standard Form knowledge point
\[

$$\begin{aligned} &y = ax^2 + bx + c\\ &y = -3x^2 - 30x - 85\\ &a = -3,\quad b = -30,\quad c = -85 \end{aligned}$$

\]

Calculate the x-coordinate of the vertex

Using the Axis of Symmetry knowledge point
\[

$$\begin{aligned} &x = -\frac{b}{2a}\\ &x = -\frac{-30}{2(-3)} = -\frac{-30}{-6} = -5 \end{aligned}$$

\]

Calculate the y-coordinate of the vertex

Substitute \(x = -5\) into the original quadratic equation:
\[

$$\begin{aligned} y &= -3(-5)^2 - 30(-5) - 85\\ y &= -3(25) + 150 - 85\\ y &= -75 + 150 - 85\\ y &= 75 - 85\\ y &= -10 \end{aligned}$$

\]

State the vertex coordinates

The coordinates of the vertex \((h, k)\) are:
\[
(h, k) = (-5, -10)
\]
</reasoning>

<answer>
Find the vertex of this quadratic function.
\(y = -3x^2 - 30x - 85\)
(<blank>-5</blank>, <blank>-10</blank>)
</answer>

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