Answer:
a) The transformed function in vertex form after the translations and horizontal shrink is g(x) = 2(x - 3)² - 4.
b) The transformed function in vertex form after the horizontal shrink and translations is h(x) = (1/2)(x - 3)² - 4.
Explanation:
a) To write the transformed function in vertex form after the translations and horizontal shrink, we can follow these steps:
1. Start with the original function f(x) = x².
2. Perform the translations: 4 units down and 3 units right. This means we subtract 4 from the y-coordinate and add 3 to the x-coordinate.
3. After the translations, we have a new function g(x) = (x - 3)² - 4.
4. Apply the horizontal shrink by a factor of 1/2. This means we multiply the x-coordinate by 2.
5. The transformed function in vertex form is g(x) = 2(x - 3)² - 4.
b) If we perform the horizontal shrink first, followed by the translations, we need to reverse the order of the operations. Here are the steps:
1. Start with the original function f(x) = x².
2. Apply the horizontal shrink by a factor of 1/2. This means we divide the x-coordinate by 2.
3. After the horizontal shrink, we have a new function h(x) = (1/2)x².
4. Perform the translations: 4 units down and 3 units right. This means we subtract 4 from the y-coordinate and add 3 to the x-coordinate.
5. The transformed function in vertex form is h(x) = (1/2)(x - 3)² - 4.
In summary:
a) The transformed function in vertex form after the translations and horizontal shrink is g(x) = 2(x - 3)² - 4.
b) The transformed function in vertex form after the horizontal shrink and translations is h(x) = (1/2)(x - 3)² - 4.