To find the transformation of the function [tex]\( f(x) = 3(x+2)^2 \)[/tex] to get the function [tex]\( g(x) \)[/tex] which results from a translation 1 unit left and 6 units down, followed by a vertical stretch by a factor of 2, we will follow these steps:
### Step-by-Step Solution:
1. Original Function:
[tex]\( f(x) = 3(x+2)^2 \)[/tex]
2. Translate the Function:
- Translating a function 1 unit left involves replacing [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex].
- Translating a function 6 units down involves subtracting 6 from the function.
Applying the translation:
[tex]\[
g_{translated}(x) = f(x + 1) - 6
\][/tex]
Let's compute [tex]\( f(x + 1) \)[/tex]:
[tex]\[
f(x + 1) = 3(x + 1 + 2)^2 = 3(x + 3)^2
\][/tex]
Now subtract 6:
[tex]\[
g_{translated}(x) = 3(x + 3)^2 - 6
\][/tex]
3. Vertical Stretch:
- A vertical stretch by a factor of 2 involves multiplying the function by 2.
Applying the vertical stretch:
[tex]\[
g(x) = 2 \cdot g_{translated}(x)
\][/tex]
Substitute [tex]\( g_{translated}(x) \)[/tex] into the equation:
[tex]\[
g(x) = 2 \left( 3(x + 3)^2 - 6 \right)
\][/tex]
4. Simplify the Expression:
Distribute the 2:
[tex]\[
g(x) = 2 \cdot 3(x + 3)^2 - 2 \cdot 6
\][/tex]
Simplify:
[tex]\[
g(x) = 6(x + 3)^2 - 12
\][/tex]
### Final Function:
[tex]\[
g(x) = 6(x + 3)^2 - 12
\][/tex]
Thus, the function [tex]\( g(x) \)[/tex] which results from a translation 1 unit left and 6 units down, followed by a vertical stretch by a factor of 2 of the graph of [tex]\( f(x) = 3(x+2)^2 \)[/tex], is:
[tex]\[
g(x) = 6(x+3)^2 - 12
\][/tex]
