graph the rational function.

\f(x) = \frac{3x - 3}{-x + 2}\

start by drawing the vertical and horizontal asymptotes. then plot two points on each piece of the graph. finally, click on the graph-a-function button.

Question

graph the rational function.

\f(x) = \frac{3x - 3}{-x + 2}\

start by drawing the vertical and horizontal asymptotes. then plot two points on each piece of the graph. finally, click on the graph-a-function button.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Asymptotes", "Graphing Rational Functions", "Rational Function Key Points" ], "new_concepts": [], "current_concepts": [ "Asymptotes", "Graphing Rational Functions", "Rational Function Key Points" ] } ## Find the asymptotes Using the *Asymptotes* and *Graphing Rational Functions* knowledge points $$ \begin{aligned} &\text{Vertical asymptote: } -x + 2 = 0 \implies x = 2\ &\text{Horizontal asymptote: } y = \frac{3}{-1} \implies y = -3 \end{aligned} $$ ## Find key points on the left piece Using the *Rational Function Key Points* knowledge point $$ \begin{aligned} &x = 1 \implies f(1) = \frac{3(1)-3}{-1+2} = 0 \implies (1, 0)\ &x = 0 \implies f(0) = \frac{3(0)-3}{-0+2} = -1.5 \implies (0, -1.5) \end{aligned} $$ ## Find key points on the right piece Using the *Rational Function Key Points* knowledge point $$ \begin{aligned} &x = 3 \implies f(3) = \frac{3(3)-3}{-3+2} = \frac{6}{-1} = -6 \implies (3, -6)\ &x = 4 \implies f(4) = \frac{3(4)-3}{-4+2} = \frac{9}{-2} = -4.5 \implies (4, -4.5) \end{aligned} $$ To graph the rational function $f(x) = \frac{3x - 3}{-x + 2}$: 1. **Asymptotes**: - **Vertical Asymptote**: $x = 2$ - **Horizontal Asymptote**: $y = -3$ 2. **Points on the left piece ($x < 2$)**: - $(1, 0)$ - $(0, -1.5)$ 3. **Points on the right piece ($x > 2$)**: - $(3, -6)$ - $(4, -4.5)$ { "elements": [ { "type": "line", "params": [[2, -10], [2, 10]], "properties": { "name": "x = 2", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "line", "params": [[-10, -3], [10, -3]], "properties": { "name": "y = -3", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "point", "params": [[1, 0]], "properties": { "name": "(1, 0)", "size": 4, "strokeColor": "#8C55F2", "fillColor": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -1.5]], "properties": { "name": "(0, -1.5)", "size": 4, "strokeColor": "#8C55F2", "fillColor": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[3, -6]], "properties": { "name": "(3, -6)", "size": 4, "strokeColor": "#8C55F2", "fillColor": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[4, -4.5]], "properties": { "name": "(4, -4.5)", "size": 4, "strokeColor": "#8C55F2", "fillColor": "#8C55F2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "(3*x - 3)/(-x + 2)", "latex": "\frac{3x-3}{-x+2}"}, -10, 1.9], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "functiongraph", "params": [{"js": "(3*x - 3)/(-x + 2)", "latex": "\frac{3x-3}{-x+2}"}, 2.1, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Graphing Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the asymptotes

Find key points on the left piece

Find key points on the right piece

Answer: