Math Problem Statement
Solution
We are given the function and need to determine its transformed images based on three transformations:
- Translation by the vector .
- Reflection about the line .
- Dilation with center at and scale factor of 4.
Let’s go through each part in detail:
a. Translation by :
A translation shifts the function horizontally by 2 units and vertically by 3 units.
For a translation , the function becomes: Substituting and , the transformed function is: Now, substituting into this expression: Adding 3: Thus, the new function after translation is .
b. Reflection about the line :
Reflection about a vertical line changes the function as follows: For reflection about , this becomes: Substituting : So, the reflected function is .
c. Dilation with center (0,0) and scale factor 4:
For a dilation with center and scale factor , the transformation is given by: For , the function becomes: Substituting : Thus, the new function after dilation is .
Summary of Results:
- a) Translated function:
- b) Reflected function:
- c) Dilated function:
Would you like more details on any step or clarification?
Here are five related questions to explore:
- How does translating a function affect its graph in general?
- Can you describe how reflection over different lines (like ) changes a function?
- What would happen if the dilation had a different center point?
- How would the transformations change if the function were quadratic, like ?
- How can you combine multiple transformations into a single operation?
Tip: Reflections and dilations around points other than the origin often involve more complex transformations, but they follow similar principles by adjusting coordinates accordingly.
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Transformation
Translation
Reflection
Dilation
Formulas
Translation formula: f(x - h) + k
Reflection formula: f(2c - x) for reflection over vertical line x = c
Dilation formula: f(x/k) for dilation with scale factor k
Theorems
Basic Transformations in Function Graphs
Suitable Grade Level
Grades 10-12
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