which of the following functions opens downward?

(f(x) = 0.5x^2 - 10x - 110)

(f(x) = -x^2 + 33x)

Question

which of the following functions opens downward?

(f(x) = 0.5x^2 - 10x - 110)

(f(x) = -x^2 + 33x)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Quadratic Functions", "Parabola Direction", "Leading Coefficient" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Parabola Direction", "Leading Coefficient" ] } ## Identify the quadratic function form Using the **Quadratic Functions** knowledge point $$ f(x) = ax^2 + bx + c $$ ## Determine the direction of opening Using the **Parabola Direction** knowledge point $$ \begin{aligned} &\text{If } a > 0, \text{ the parabola opens upward.}\ &\text{If } a < 0, \text{ the parabola opens downward.} \end{aligned} $$ ## Analyze the leading coefficients Using the **Leading Coefficient** knowledge point $$ \begin{aligned} &\text{For } f(x) = 0.5x^2 - 10x - 110: \quad a = 0.5 > 0 \quad (\text{opens upward})\ &\text{For } f(x) = -x^2 + 33x: \quad a = -1 < 0 \quad (\text{opens downward}) \end{aligned} $$

$f(x) = 0.5x^2 - 10x - 110$

$f(x) = -x^2 + 33x$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the quadratic function form

Determine the direction of opening

Analyze the leading coefficients

Answer: