use a transformation of the graph of $y = x^4$ to graph the function.

\h(x) = 2 - (x - 3)^4\

choose the correct graph.

Question

use a transformation of the graph of $y = x^4$ to graph the function.

\h(x) = 2 - (x - 3)^4\

choose the correct graph.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Function Transformations" ], "new_concepts": [], "current_concepts": [ "Function Transformations", "Cubic Function Graphs" ] } ## Identify the base function and transformations $$ \begin{aligned} f(x) &= x^4 \ h(x) &= -(x - 3)^4 + 2 \end{aligned} $$ ## Determine the sequence of transformations $$ \begin{aligned} \text{Horizontal shift:} \quad & x \to x - 3 \quad \text{(shift right by 3 units)} \ \text{Reflection:} \quad & y \to -y \quad \text{(reflect across the x-axis)} \ \text{Vertical shift:} \quad & y \to y + 2 \quad \text{(shift up by 2 units)} \end{aligned} $$ ## Find key points on the transformed graph $$ \begin{aligned} (0, 0) \to (3, 0) \to (3, 0) \to (3, 2) \ (1, 1) \to (4, 1) \to (4, -1) \to (4, 1) \ (-1, 1) \to (2, 1) \to (2, -1) \to (2, 1) \end{aligned} $$ To graph the function $h(x) = 2 - (x - 3)^4$, start with the graph of $y = x^4$ and apply the following transformations: 1. Shift the graph horizontally to the right by 3 units. 2. Reflect the graph vertically across the $x$-axis. 3. Shift the graph vertically upward by 2 units. The vertex (turning point) of the transformed graph is located at $(3, 2)$, and the curve opens downward. { "elements": [ { "type": "functiongraph", "params": [ { "js": "2 - Math.pow(x - 3, 4)", "latex": "2 - (x - 3)^4" }, 0, 6 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "h(x)" } }, { "type": "point", "params": [ [3, 2] ], "properties": { "name": "(3, 2)", "size": 4, "color": "#F2557F", "strokeColor": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [2, 1] ], "properties": { "name": "(2, 1)", "size": 3, "color": "#5583F2", "strokeColor": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [4, 1] ], "properties": { "name": "(4, 1)", "size": 3, "color": "#5583F2", "strokeColor": "#5583F2", "withLabel": true } } ], "timestamps": [] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Function Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the base function and transformations

Determine the sequence of transformations

Find key points on the transformed graph

Answer: