Respuesta :
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:
[tex]y= (x-1)^2-3[/tex]
Now, this graph is translated to get a new graph as:
[tex]y=\dfrac{1}{2}(x+4)^2[/tex]
- 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.
[tex]y=(x-1)^2-3+3\\\\i.e.\\\\y=(x-1)^2[/tex]
- 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,
[tex]y=(x-1+5)^2\\\\i.e.\\\\y=(x+4)^2[/tex]
- 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.
[tex]y=\dfrac{1}{2}(x+4)^2\\\\i.e.\\\\k=\dfrac{1}{2}<1[/tex]
To solve the problem we must know about the equation of a parabola.
When the graph of y= (x-1)^2-3 is transformed to produce the graph of y= 1/2(x+4)^2, the vertex of the parabola shifts from (1,-3) to (-4,0) and the graph of the parabola gets wider as well by a factor of 2.
What is the Equation of a parabola?
[tex]y = a(x-h)^2 + k[/tex]
where,
(h, k) are the coordinates of the vertex of the parabola in form (x, y);
a defines how narrower is the parabola, If the value of a is more the graph will be narrower, and the "-" or "+" that the parabola will open up or down.
Given to us
[tex]y= (x-1)^2-3[/tex]
[tex]y= \dfrac{1}{2}(x+4)^2[/tex]
As we have revised the concept of a parabola, if we compare the two given equations with the general equation of a parabola,
In equation 1,
[tex]y= (x-1)^2-3[/tex]
[tex]y = a(x-h)^2 + k[/tex]
(h,k) = (1,-3)
a = +1
In equation 2,
[tex]y= \dfrac{1}{2}(x+4)^2[/tex]
[tex]y = a(x-h)^2 + k[/tex]
(h,k) = (-4,0)
a = +1/2
- As we can see in the above comparison the coordinate of the vertex of the parabola changes from (1,-3) to (-4,0).
- Also, the value of 'a' changes from 1 to 1/2, therefore, the graph gets wider by a factor of 2.
The above conclusions can be seen in the graph plotted below.
Hence, when the graph of y= (x-1)^2-3 is transformed to produce the graph of y= 1/2(x+4)^2, the vertex of the parabola shifts from (1,-3) to (-4,0) and the graph of the parabola gets wider as well by a factor of 2.
Learn more about Parabola:
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