Math Problem Statement
Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. g left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis cubed plus 6 Question content area bottom left Part 1 Use the graphing tool to graph the equation.
Solution
The given function is . We need to analyze how this function can be graphed using techniques like shifting, compressing, stretching, and reflecting, and then find the domain and range.
Step-by-Step Graphing Process:
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Start with the basic function: The basic function is . This is a cubic function with its characteristic shape (an "S"-like curve that passes through the origin). The domain of is , and its range is also .
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Horizontal Shift: The function has been modified to . The inside the cube indicates a horizontal shift. Specifically, the graph of is shifted 1 unit to the left. This means that instead of the curve passing through , it now passes through .
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Vertical Shift: The outside the cubic term indicates a vertical shift. The graph is shifted 6 units upward, so the point moves to .
Thus, the graph of is the graph of shifted 1 unit left and 6 units up.
Domain and Range:
- Domain: Since this is a cubic function, the domain remains as the function is defined for all real values of .
- Range: The cubic function has a range of , and since there is only a vertical shift by 6 units, the range of also remains .
Summary:
- Horizontal shift: 1 unit to the left
- Vertical shift: 6 units upward
- Domain:
- Range:
If you would like, I can help you graph the function visually or provide more detailed steps.
Would you like any details or further clarification?
Here are 5 questions to expand your understanding:
- How does a horizontal shift affect the function's behavior?
- What happens to the range if a cubic function is vertically stretched?
- Can the domain of a cubic function ever be restricted by transformations?
- How would the graph change if the function was instead?
- What role does the coefficient of play in stretching or compressing the cubic graph?
Tip: Horizontal shifts move the graph left or right, depending on whether the shift is positive or negative.
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Math Problem Analysis
Mathematical Concepts
Algebra
Graphing Transformations
Cubic Functions
Formulas
g(x) = (x + 1)^3 + 6
Theorems
Transformations of Functions
Domain and Range Theorems
Suitable Grade Level
Grades 10-12
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