3). graph the quadratic function.
$f(x) = -2(x + 1)^2 + 8$ by finding the following.
(a). determine whether the graph opens upward or downward.
(b). state whether a maximum or minimum point occurs at the vertex of the parabola.
(c). vertex, & axis of symmetry.
(d). y-intercept & x-intercepts.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Graphing Parabolas", "y-intercept of a Graph", "Quadratic Intercepts" ], "new_concepts": [], "current_concepts": [ "Graphing Parabolas", "y-intercept of a Graph", "Quadratic Intercepts", "Vertex Form of a Quadratic" ] } ## Determine orientation and extremum type $$ \begin{aligned} &f(x) = a(x-h)^2 + k \implies f(x) = -2(x+1)^2 + 8 \ &a = -2 < 0 \implies \text{opens downward} \ &a < 0 \implies \text{maximum point occurs at the vertex} \end{aligned} $$ ## Find vertex and axis of symmetry $$ \begin{aligned} &h = -1, \quad k = 8 \implies \text{Vertex: } (-1, 8) \ &\text{Axis of symmetry: } x = -1 \end{aligned} $$ ## Calculate intercepts $$ \begin{aligned} &\text{y-intercept: } f(0) = -2(0+1)^2 + 8 = 6 \implies (0, 6) \ &\text{x-intercepts: } -2(x+1)^2 + 8 = 0 \ &(x+1)^2 = 4 \implies x+1 = \pm 2 \ &x = 1 \text{ or } x = -3 \implies (1, 0), (-3, 0) \end{aligned} $$ (a) Downward (b) Maximum (c) Vertex: $(-1, 8)$, Axis of symmetry: $x = -1$ (d) Y-intercept: $(0, 6)$, X-intercepts: $(1, 0)$ and $(-3, 0)$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "-2*Math.pow(x+1, 2) + 8", "latex": "-2(x+1)^2 + 8" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "point", "params": [ [-1, 8] ], "properties": { "name": "Vertex (-1,8)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [0, 6] ], "properties": { "name": "(0,6)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [1, 0] ], "properties": { "name": "(1,0)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [-3, 0] ], "properties": { "name": "(-3,0)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "line", "params": [ [-1, -10], [-1, 10] ], "properties": { "strokeColor": "#583C87", "strokeWidth": 1, "dash": 2, "name": "x = -1", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Parabolas" ] }

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Explanation:

Determine orientation and extremum type

Find vertex and axis of symmetry

Calculate intercepts

Answer: