find the range of this quadratic function.

\y = x^2 - 2x - 1\

\-2 \le y

Question

find the range of this quadratic function.

\y = x^2 - 2x - 1\

\-2 \le y < \infty\

\-\infty < y < \infty\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Functions", "Parabola Vertex Identification", "Range of a Function" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Parabola Vertex Identification", "Range of a Function" ] } ## Find the vertex of the parabola $$ \begin{aligned} x_v &= -\frac{b}{2a} = -\frac{-2}{2(1)} = 1\ y_v &= (1)^2 - 2(1) - 1 = -2 \end{aligned} $$ ## Determine the direction and range $$ \begin{aligned} a &= 1 > 0 \quad \text{(opens upward)}\ y &\ge -2 \implies -2 \le y < \infty \end{aligned} $$

(A) $-2 \le y < \infty$

(B) $-\infty < y < \infty$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Range of a Function" ] }

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QUESTION IMAGE

Question

Explanation:

Find the vertex of the parabola

Determine the direction and range

Answer: