Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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]
### 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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.