graph the function.

\
(x) = \begin{cases} -x^2 + 1 & \text{for } x \le 1 \\ 5x - 5 & \text{for } x 1 \end{cases}\

part 1 of 5
the first rule defines a parabola with vertex $(0, 1)$.

part 2 of 5
since $x \le 1$, the select point $(\quad, \quad)$ is graphed as a solid dot.

Question

graph the function.

\
(x) = \begin{cases} -x^2 + 1 & \text{for } x \le 1 \\ 5x - 5 & \text{for } x > 1 \end{cases}\

part 1 of 5
the first rule defines a parabola with vertex $(0, 1)$.

part 2 of 5
since $x \le 1$, the select point $(\quad, \quad)$ is graphed as a solid dot.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Piecewise Functions", "Parabola Vertex", "Ordered Pairs" ], "new_concepts": [], "current_concepts": [ "Piecewise Functions", "Parabola Vertex", "Ordered Pairs", "Graphing Piecewise Functions" ] } ## Identify the boundary point for the first rule Using the **Piecewise Functions** and **Ordered Pairs** knowledge points $$ \begin{aligned} &\text{First rule: } y = -x^2 + 1 \quad \text{for } x \le 1\ &\text{Boundary value: } x = 1\ &y = -(1)^2 + 1 = 0\ &\text{Boundary point: } (1, 0) \end{aligned} $$ ## Determine the endpoint style Using the **Piecewise Functions** knowledge point $$ \begin{aligned} &\text{Inequality: } x \le 1 \quad (\text{includes equality})\ &\text{Endpoint type: } \text{endpoint}\ &\text{Visual representation: } \text{solid dot} \end{aligned} $$ Since $x \le 1$, the endpoint point $(1, 0)$ is graphed as a solid dot. { "elements": [ { "type": "functiongraph", "params": [ { "js": "-x*x + 1", "latex": "-x^2 + 1" }, -10, 1 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "functiongraph", "params": [ { "js": "5*x - 5", "latex": "5x - 5" }, 1, 10 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3, "dash": 2 } }, { "type": "point", "params": [ [0, 1] ], "properties": { "name": "Vertex (0,1)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [ [1, 0] ], "properties": { "name": "(1,0)", "size": 5, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [ [1, 0] ], "properties": { "size": 5, "color": "white", "strokeColor": "#5583F2", "strokeWidth": 2, "withLabel": false } } ], "timestamps": [ 0.5, 1.0, 1.5 ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Piecewise Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the boundary point for the first rule

Determine the endpoint style

Answer: