Math Problem Statement
Solution
The given problem asks for the transformation of the function into a new function , involving three transformations:
- Vertical shrink by a factor of .
- Reflection in the y-axis.
- Translation 3 units to the right.
Let's break down the transformation step by step:
Step 1: Vertical Shrink by
A vertical shrink multiplies the entire function by , so the function becomes:
Step 2: Reflection in the y-axis
A reflection in the y-axis replaces with . Applying this reflection: Note that reflecting about the y-axis doesn’t change the function because .
Step 3: Translation 3 units to the right
To translate the function 3 units to the right, replace with . The equation now becomes:
Vertex
The vertex of is at . After translating 3 units to the right, the new vertex is at .
Final Answer
The rule for is: The vertex is .
Would you like further details on any of the steps?
Here are 5 related questions to extend your understanding:
- How does a vertical shrink differ from a vertical stretch?
- What is the effect of translating a function vertically instead of horizontally?
- How do you find the vertex of any quadratic function in the form ?
- What happens to the graph of when reflected over the x-axis instead of the y-axis?
- Can a function be reflected over both axes simultaneously?
Tip: When reflecting over the y-axis, quadratic functions often remain unchanged because .
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Math Problem Analysis
Mathematical Concepts
Quadratic Transformations
Function Reflection
Function Translation
Formulas
g(x) = a(x - h)^2 + k
Theorems
Transformation Theorems for Quadratic Functions
Suitable Grade Level
Grades 9-11