Math Problem Statement
The graph of y = (x + 1)^2 is transformed by the composition of two transformations to the graph of y = 2(x - 4)^2. Find these two transformations.
Solution
To transform the graph of into , we need to identify two transformations applied to the original graph. These transformations include changes in position (horizontal and vertical shifts) and scaling (vertical stretching or compressing).
Step 1: Transformation from to
This involves a horizontal shift:
- The expression becomes .
- This indicates a shift 7 units to the right because:
Step 2: Transformation from to
This involves a vertical scaling:
- The coefficient of the quadratic term changes from to .
- This represents a vertical stretch by a factor of 2, meaning the graph becomes twice as steep.
Final Answer:
- Horizontal shift by 7 units to the right.
- Vertical stretch by a factor of 2.
Verification:
- Starting from , shifting it 7 units to the right gives .
- Scaling this vertically by a factor of 2 gives .
Do you have any further questions or want a detailed explanation?
Related Questions:
- What would the transformations be if were transformed to ?
- How do you graphically represent a vertical stretch on a parabola?
- If the transformations were reversed, how would transform back to ?
- What happens to the vertex of after these transformations?
- How can horizontal shifts and vertical stretches be combined into a single transformation matrix?
Tip: When dealing with transformations, always track the changes in the vertex and the shape of the graph as a starting point!
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Math Problem Analysis
Mathematical Concepts
Transformations of functions
Quadratic functions
Formulas
y = a(x - h)^2 + k (vertex form of a quadratic function)
Theorems
Transformation properties of quadratic graphs
Suitable Grade Level
Grades 9-12