4). graph the quadratic function.
$f(x) = x^2 + 2x - 8$ 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", "Quadratic Intercepts", "Factoring Quadratics" ], "new_concepts": [], "current_concepts": [ "Graphing Parabolas", "Quadratic Intercepts", "Factoring Quadratics", "Vertex of a Parabola", "Axis of Symmetry" ] } ## Analyze direction of opening and extremum type $$ \begin{aligned} &f(x) = x^2 + 2x - 8 \ &a = 1 > 0 \implies \text{opens upward} \ &a > 0 \implies \text{minimum point occurs at the vertex} \end{aligned} $$ ## Calculate vertex and axis of symmetry $$ \begin{aligned} &x_v = -\frac{b}{2a} = -\frac{2}{2(1)} = -1 \ &y_v = f(-1) = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9 \ &\text{Vertex: } (-1, -9) \ &\text{Axis of symmetry: } x = -1 \end{aligned} $$ ## Determine y-intercept and x-intercepts $$ \begin{aligned} &y\text{-intercept: } f(0) = -8 \implies (0, -8) \ &x^2 + 2x - 8 = 0 \implies (x+4)(x-2) = 0 \ &x\text{-intercepts: } (-4, 0) \text{ and } (2, 0) \end{aligned} $$ (a) The graph opens upward. (b) A minimum point occurs at the vertex. (c) Vertex: $(-1, -9)$; Axis of symmetry: $x = -1$ (d) Y-intercept: $(0, -8)$; X-intercepts: $(-4, 0)$ and $(2, 0)$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "x*x + 2*x - 8", "latex": "x^2 + 2x - 8" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "point", "params": [ [-1, -9] ], "properties": { "name": "Vertex (-1, -9)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [-4, 0] ], "properties": { "name": "(-4, 0)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [2, 0] ], "properties": { "name": "(2, 0)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [0, -8] ], "properties": { "name": "(0, -8)", "size": 4, "color": "#55DDF2", "withLabel": true } }, { "type": "line", "params": [ [-1, -10], [-1, 10] ], "properties": { "name": "x = -1", "strokeColor": "#F28A55", "strokeWidth": 1.5, "dash": 2, "withLabel": true } } ], "timestamps": [] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Parabolas" ] }

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Analyze direction of opening and extremum type

Calculate vertex and axis of symmetry

Determine y-intercept and x-intercepts

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