Question

describe the transformations of the parent function $f(x)=x^{2}$ used to get the new function $g(x)=(x - 3)^{2}-5$.
translation 3 units to the left and 5 units up
translation 3 units to the left and 5 units down
translation 3 units to the right and 5 units up
translation 3 units to the right and 5 units down

Explanation:

Step1: Analyze horizontal shift

For a function of the form $y=(x - h)^2 + k$ compared to the parent function $y = x^2$, when $h>0$, the graph shifts to the right. In $g(x)=(x - 3)^2-5$, $h = 3$, so there is a translation 3 units to the right.

Step2: Analyze vertical shift

For the function $y=(x - h)^2 + k$, when $k<0$, the graph shifts down. In $g(x)=(x - 3)^2-5$, $k=- 5$, so there is a translation 5 units down.

Answer:

Translation 3 units to the right and 5 units down