find the vertex of this quadratic function.

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

(?, )

Question

find the vertex of this quadratic function.

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

(?,  )

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Axis of Symmetry", "Quadratic Standard Form" ], "new_concepts": [ "Vertex of a Quadratic Function", "Vertex Formula" ], "current_concepts": [ "Quadratic Standard Form", "Axis of Symmetry", "Vertex of a Quadratic Function" ] } ## 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: $$ (h, k) = (-5, -10) $$ Find the vertex of this quadratic function. $y = -3x^2 - 30x - 85$ (-5, -10) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Vertex of a Quadratic Function" ] }

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Question

Explanation:

Identify coefficients from standard form

Calculate the x-coordinate of the vertex

Calculate the y-coordinate of the vertex

State the vertex coordinates

Answer: