To determine the quadratic function that is obtained from the parent function $f (x) = x^{2}$ after the given sequence of transformations, let's consider the transformations one at a time.

Vertical Shrink by a Factor of $\frac{1}{3}$

We will start by performing a vertical shrink on the parent function by a factor of $\frac{1}{3} .$ This is done by multiplying the parent function by $\frac{1}{3} .$ The resulting function is $y = \frac{1}{3} x^{2} .$

Reflection in the $y -$ axis

Next, let's reflect the function across the

y -

axis by multiplying the variable by

- 1 .

y = \frac{1}{3} (- x)^{2} \Leftrightarrow y = \frac{1}{3} x^{2}

Notice, in this case the function does not change because

(- x)^{2} = x^{2} .

Horizontal Translation $3$ Units Right

Finally, to perform a horizontal translation $4$ units right we need to subtract $3$ from the $x -$ variable. The result is $g (x) = \frac{1}{3} (x - 3)^{2} .$

The quadratic function that is obtained after the sequence of transformations is $g (x) = \frac{1}{3} (x - 2)^{2} .$

Vertex

The vertex of the quadratic parent function is $(0, 0) .$ We can find the vertex of $g$ using the transformations found above.

Transformations	Change in coordinates
Vertical shrink	$(x, \frac{1}{3} y)$
Reflection in the $y -$ axis	$(- x, \frac{1}{3} y)$
Horizontal translation right	$(- x + 3, \frac{1}{3} y)$

Let's substitute the coordinates of a parent function to find the new vertex.

(- x + 3, \frac{1}{3} y)

SubstituteII

$x = 0$ , $y = 0$

(- 0 + 3, \frac{1}{3} (0))

ZeroPropMult

Zero Property of Multiplication

(- 0 + 3, 0)

AddTerms

Add terms

(3, 0)

The vertex of

g (x)

is

(3, 0) .

Vertical Shrink by a Factor of 31​

Reflection in the y-axis

Horizontal Translation 3 Units Right

Vertex

Vertical Shrink by a Factor of $\frac{1}{3}$

Reflection in the $y -$ axis

Horizontal Translation $3$ Units Right

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