Answer: g(x) = (1/3)*(-x)^2 + 3.
Step-by-step explanation:
Let's define all the transformations in a general way,
A vertical contraction/dilation of a scale factor a is written as:
g(x) = a*f(x)
where if a > 1, it is a dilation.
if 0 < a < 1, it is a contraction.
A reflection in the y-axis changes the value of the x-component and leaves the y-component invariable, then this transformation is written as:
g(x) = f(-x).
A translation of A units up (A positive) is written as:
g(x) = f(x) + A.
Then for this case we have:
A dilation with scale factor 1/3.
g(x) = (1/3)*f(x)
A reflection over the y-axis.
g(x) = (1/3)*f(-x)
A translation of 3 units up.
g(x) = (1/3)*f(-x) + 3
(in that order, remember that the order in which you applly the transformations matters)
Then g(x) = (1/3)*f(-x) + 3
and f(x) = x^2, so we can replace that in the equation for g(x)
g(x) = (1/3)*(-x)^2 + 3.
Transformation involves changing the position of a function.
The rule for g is: [tex]\mathbf{g(x) = \frac 13(x - 3)^2}[/tex]
The parent function is given as:
[tex]\mathbf{f(x) = x^2}[/tex]
The rule of vertical shrink by a factor of 1/3 is:
[tex]\mathbf{(x,y) \to (x,\frac 13y)}[/tex]
So, we have:
[tex]\mathbf{f'(x) = \frac 13x^2}[/tex]
The rule of reflection across the y-axis is:
[tex]\mathbf{(x,y) \to (-x,y)}[/tex]
So, we have:
[tex]\mathbf{f''(x) = \frac 13(-x)^2}[/tex]
[tex]\mathbf{f''(x) = \frac 13x^2}[/tex]
The rule of translation 3 units right is:
[tex]\mathbf{(x,y) \to (x - 3,y)}[/tex]
So, we have:
[tex]\mathbf{g(x) = \frac 13(x - 3)^2}[/tex]
Hence, the rule for g is: [tex]\mathbf{g(x) = \frac 13(x - 3)^2}[/tex]
Read more about transformations at:
https://brainly.com/question/11709244
