Math Problem Statement
Solution
We are given two functions:
Analysis of transformations:
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Starting with : This is the basic cubic root function , which we can assume is plotted normally.
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Transformation for :
- represents a transformation of :
- The term means a horizontal shift left by 4 units.
- The means a vertical shift down by 5 units.
- represents a transformation of :
Now, we analyze the given options:
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First option: "The graph of is a translation 5 units left and 4 units up" is incorrect since the horizontal shift is 4 units left, not 5, and there is no vertical translation up.
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Second option: "The graph of is a translation 4 units left and 5 units down" is correct based on the transformation observed.
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Third option: "The graph of is a translation 5 units left and 4 units down" is incorrect for the same reasons as the first option.
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Fourth option: "The graph of is a translation 4 units right and 5 units down" is incorrect because the shift is to the left, not right.
Thus, the correct description is the second option.
Choosing the correct graph:
We need to identify which graph matches the transformations: should be shifted 4 units left and 5 units down compared to .
Would you like me to go into detail regarding the graphs, or have any specific questions on transformations?
Here are five related questions:
- How do we determine if a function is shifted horizontally or vertically?
- What is the effect of adding or subtracting constants inside the function versus outside?
- How would the transformation change if we had ?
- What is the general shape of a cubic root function?
- How do we confirm the direction of shifts by inspecting the equation?
Tip: For horizontal transformations, focus on changes inside the function's argument. A shift left occurs when the constant is added, and a shift right when it's subtracted.
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Math Problem Analysis
Mathematical Concepts
Transformations of functions
Cubic root functions
Formulas
g(x) = ∛(x + 4) - 5
Theorems
Transformation theorem: Horizontal and vertical shifts
Suitable Grade Level
Grades 9-12