Question

How is the graph of y= (x-1)^2-3 transformed to produce the graph of y= 1/2(x+4)^2?

Answer 1

Answer:

The transformation that follows are:

• Shift 2 units up.

• Shift 5 units to the left.

• Vertical shrink by a factor of 2.

Step-by-step explanation:

We are given a graph as:

     $y= (x-1)^2-3$

Now, this graph is translated to get a new graph as:

       $y=\dfrac{1}{2}(x+4)^2$

• First the graph of the original function is shifted 3 units up.

Since, the translation of the type:

f(x) to f(x)+k is a shift k units up or k units down depending on whether k is positive or negative respectively.

$y=(x-1)^2-3+3\\\\i.e.\\\\y=(x-1)^2$

• Now, this graph is then shifted 5 units left.

since the translation of the type f(x) to f(x+k) is a shift of  the function f(x) k units to the left or k units to the right depending on k whether k is positive or negative respectively.

Here,

$y=(x-1+5)^2\\\\i.e.\\\\y=(x+4)^2$

• Now, this graph is then vertically shrink by a factor of 2.

Because the transformation of the type:

         f(x) to kf(x) is a vertical stretch if k>1

and vertical shrink if 0<k<1.

i.e.

$y=\dfrac{1}{2}(x+4)^2\\\\i.e.\\\\k=\dfrac{1}{2}$

Answer 2

Answer:

Original graph

y = (x-1)^2 - 3

Transformed graph

y = 1/2 (x+4)^2

From the original graph:

y = (x-1)^2 - 3

The function is shifted upward 3 units

y = (x-1)^2 -3 +3

y = (x-1)^2

Then the function is shifted 5 units to the left

y = (x-1+5)^2

y= (x+4)^2

The graph will be stretched vertically with a factor of 1/2

y = 1/2 (x+4)^2 ..