To derive the function [tex]\( g(x) \)[/tex] from the given function [tex]\( f(x) = 3(x + 2)^2 \)[/tex] through the specified transformations, follow these steps:
### 1. Translation 1 Unit Left
When you translate a function [tex]\( f(x) \)[/tex] by 1 unit to the left, you replace [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex] in the function. Start with:
[tex]\[ f(x) = 3(x + 2)^2 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex]:
[tex]\[ g_1(x) = 3((x + 1) + 2)^2 \][/tex]
[tex]\[ g_1(x) = 3(x + 3)^2 \][/tex]
### 2. Translation 6 Units Down
Translating a function down by 6 units means subtracting 6 from the function [tex]\( g_1(x) \)[/tex]:
[tex]\[ g_2(x) = g_1(x) - 6 \][/tex]
[tex]\[ g_2(x) = 3(x + 3)^2 - 6 \][/tex]
### 3. Vertical Stretch by a Factor of 2
Applying a vertical stretch by a factor of 2 means multiplying the entire function [tex]\( g_2(x) \)[/tex] by 2:
[tex]\[ g(x) = 2 \cdot g_2(x) \][/tex]
[tex]\[ g(x) = 2 \cdot (3(x + 3)^2 - 6) \][/tex]
Distribute the 2:
[tex]\[ g(x) = 2 \cdot 3(x + 3)^2 - 2 \cdot 6 \][/tex]
[tex]\[ g(x) = 6(x + 3)^2 - 12 \][/tex]
Thus, the function [tex]\( g(x) \)[/tex], after translating 1 unit left, 6 units down, and stretching vertically by a factor of 2, is:
[tex]\[ g(x) = 6(x + 3)^2 - 12 \][/tex]
