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 \(\rightarrow\) f(x-a) -b
Given the function: \(y = \frac{1}{2}(x+4)^2\)
Now, replace y with \(2 y\) results in a vertical stretch by a factor of 2.
⇒ \(y = (x+4)^2\)
Then, by definition of above;
now, shift function \(y = (x+4)^2\) as 5 units right and 3 units down we have;
\(y = (x+4-5)^2 -3\) or
\(y = (x-1)^2 -3\)
Therefore, the transformation from the graph \(y = \frac{1}{2}(x+4)^2\) is vertically stretch by a factor 2 and it is shift to 5 units right and 3 units down we have then, \(y = (x-1)^2 -3\)
Also, you can see the graph shown below;
