Using translation concepts, it is found that the equations that represents the transformations formed by the following items are given by:
- a) [tex]g(x) = \frac{1}{3}\sqrt{x + 4}[/tex]
- b) [tex]g(x) = 4\sqrt{3} + 3[/tex]
- c) [tex]g(x) = \sqrt{\frac{x}{2}} - 3[/tex]
What is a translation?
- A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
Item a:
The function is:
[tex]f(x) = \sqrt{x}[/tex]
Horizontal shift of 4 units to the left, hence:
[tex]g(x) = f(x + 4) = \sqrt{x + 4}[/tex]
Vertical compression by a factor of [tex]\frac{1}{3}[/tex], hence a multiplication by [tex]\frac{1}{3}[/tex], that is:
[tex]g(x) = \frac{1}{3}\sqrt{x + 4}[/tex]
Item b:
Vertical stretch by a factor of 4, that is, a multiplication by 4, so:
[tex]4f(x) = 4\sqrt{3}[/tex]
Vertical shift of 3 units up, hence addition of 3, that is:
[tex]g(x) = 4\sqrt{3} + 3[/tex]
Item c:
Horizontal stretch by a factor of 2, that is:
[tex]g(x) = f(\frac{1}{2}x) = \sqrt{\frac{x}{2}}[/tex]
Vertical shift of 3 units down, hence subtraction by 3, that is:
[tex]g(x) = \sqrt{\frac{x}{2}} - 3[/tex]
You can learn more about translation concepts at https://brainly.com/question/4521517
