Math Problem Statement
y=2g(x+4) graph
Solution
To graph the function , follow these steps:
1. Understand the Transformations:
The function represents a transformation of the base function .
- Horizontal Shift: The term shifts the graph of horizontally to the left by 4 units.
- Vertical Stretch: The coefficient 2 vertically stretches the graph, making it twice as tall as the original graph of .
2. Start with the Base Graph:
Begin by sketching the graph of the base function . Since is not specified, treat it as a generic function.
3. Apply the Transformations:
- Horizontal Shift: Shift the graph of 4 units to the left. This means every point on the graph of moves to .
- Vertical Stretch: Multiply the -values of the shifted graph by 2. This means if a point was on the graph after the horizontal shift, it now moves to .
4. Sketch the Final Graph:
Combine the transformations to produce the graph of .
Example:
If is (a basic square root function):
- Base Graph: The graph of starts at and increases slowly, curving upwards.
- Horizontal Shift: Shifting it left by 4 units, the graph starts at .
- Vertical Stretch: Multiply all -values by 2, so the graph becomes steeper, starting at but increasing more rapidly.
Would you like to see the graph for a specific base function or have any questions?
Here are some follow-up questions:
- What function would you like to use for ?
- How does a vertical stretch affect the graph of a function?
- Can you explain the difference between a horizontal shift and a vertical shift?
- How would the graph change if the function were ?
- What if we had instead?
- How do you apply multiple transformations in sequence?
- What does the graph look like if is a quadratic function?
- What impact does a negative coefficient (e.g., ) have on the graph?
Tip: When multiple transformations are applied to a graph, it's important to perform horizontal shifts before applying vertical stretches or compressions.
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Math Problem Analysis
Mathematical Concepts
Functions
Transformations
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12