Answer:
Vertical stretch and compression:
Given a function f(x) , a new function g(x) =a f(x), where a is the constant, is a vertical stretch or vertical compression of the function f(x).
- if 0<a<1 , then the graph will be compressed
- if a > 1 the graph will be stretched .
- if a<0, then there will be combination of a vertical stretch or compression with a vertical reflection.
In general if a function is shifted a units right and b units down we can summaries this as:
f(x) is shifted a units right and b units down [tex]\rightarrow[/tex] f(x-a) -b
Given the function: [tex]y = \frac{1}{2}(x+4)^2[/tex]
Now, replace y with [tex]2 y[/tex] results in a vertical stretch by a factor of 2.
⇒ [tex]y = (x+4)^2[/tex]
Then, by definition of above;
now, shift function [tex]y = (x+4)^2[/tex] as 5 units right and 3 units down we have;
[tex]y = (x+4-5)^2 -3[/tex] or
[tex]y = (x-1)^2 -3[/tex]
Therefore, the transformation from the graph [tex]y = \frac{1}{2}(x+4)^2[/tex] is vertically stretch by a factor 2 and it is shift to 5 units right and 3 units down we have then, [tex]y = (x-1)^2 -3[/tex]
Also, you can see the graph shown below;
