To identify the parent function for the given function [tex]\( f(x) = 3(x+2)^2 \)[/tex], we need to understand the components and the structure of the expression.
1. Examine the Given Function:
The function is [tex]\( f(x) = 3(x+2)^2 \)[/tex].
- This function is a quadratic function because it involves [tex]\( (x + 2)^2 \)[/tex], which is a squared term.
2. Simplify the Function:
Let’s rewrite the function in a more recognizable form:
[tex]\( f(x) = 3(x+2)^2 \)[/tex].
3. Identify Transformations:
- Shift: The term [tex]\( (x+2) \)[/tex] indicates a horizontal shift to the left by 2 units.
- Vertical Stretch: The coefficient 3 indicates a vertical stretch by a factor of 3.
These transformations are applied to the basic quadratic function [tex]\( y = x^2 \)[/tex].
4. Determine the Parent Function:
The parent function is the simplest form of the given function without any transformations.
Given [tex]\( f(x) = 3(x+2)^2 \)[/tex]:
- Removing the horizontal shift (by ignoring the +2 inside the parentheses),
- Ignoring the vertical stretch factor (3).
The resulting basic form of the function is [tex]\( y = x^2 \)[/tex].
Therefore, the parent function of [tex]\( f(x) = 3(x+2)^2 \)[/tex] is:
A. [tex]\( y = x^2 \)[/tex]
