if $f(x) = -x^2 + 3x + 5$ and $g(x) = x^2 + 2x$, which graph shows the graph of $(f+g)(x)$?

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Linear Modeling", "Slope-Intercept Form" ], "new_concepts": [ "Function Addition", "Graphing Linear Functions" ], "current_concepts": [ "Function Addition", "Graphing Linear Functions", "Linear Modeling", "Slope-Intercept Form" ] } ## Find the sum function Using the *Linear Modeling* knowledge point $$ \begin{aligned} (f+g)(x) &= f(x) + g(x) \ &= (-x^2 + 3x + 5) + (x^2 + 2x) \ &= 5x + 5 \end{aligned} $$ ## Analyze the resulting graph Using the *Slope-Intercept Form* knowledge point $$ \begin{aligned} &y = 5x + 5 \ &\text{Slope } (m) = 5 \ &\text{y-intercept } (b) = 5 \implies (0, 5) \ &\text{x-intercept: } 5x + 5 = 0 \implies x = -1 \implies (-1, 0) \end{aligned} $$ ## Identify the correct graph The first graph shown in the image is a straight line passing through the y-intercept $(0, 5)$ and the x-intercept $(-1, 0)$. The second graph is a parabola opening downwards. Therefore, the first graph represents the function $(f+g)(x) = 5x + 5$.

(A) The first graph (a straight line with a y-intercept of 5 and an x-intercept of -1)

(B) The second graph (a downward-opening parabola)

{ "elements": [ { "type": "line", "params": [[-1, 0], [0, 5]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f+g", "withLabel": true } }, { "type": "point", "params": [[0, 5]], "properties": { "name": "(0, 5)", "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-1, 0]], "properties": { "name": "(-1, 0)", "color": "#F2557F", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Function Operations" ] }

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QUESTION IMAGE

Question

Explanation:

Find the sum function

Analyze the resulting graph

Identify the correct graph

Answer: