use transformations of the graph of ( f(x)=x^{2} ) to determine the graph of the given function.\n( g(x)=(x…

use transformations of the graph of ( f(x)=x^{2} ) to determine the graph of the given function.\n( g(x)=(x + 2)^{2} )\nthe graph of ( f(x)=x^{2} ) should be shifted 2 units to the left.\nc. the graph of ( f(x)=x^{2} ) should be shifted 2 units up.\nd. the graph of ( f(x)=x^{2} ) should be shifted 2 units to the right.\nuse the graphing tool to graph the function.\nclick to enlarge graph

use transformations of the graph of ( f(x)=x^{2} ) to determine the graph of the given function.\n( g(x)=(x + 2)^{2} )\nthe graph of ( f(x)=x^{2} ) should be shifted 2 units to the left.\nc. the graph of ( f(x)=x^{2} ) should be shifted 2 units up.\nd. the graph of ( f(x)=x^{2} ) should be shifted 2 units to the right.\nuse the graphing tool to graph the function.\nclick to enlarge graph

Answer

Explanation:

Step1: Recall the horizontal shift rule

For a function (y = f(x + h)), if (h>0), the graph of (y = f(x)) is shifted (h) units to the left. For the function (g(x)=(x + 2)^{2}), we can compare it with the parent function (f(x)=x^{2}). Here, (h = 2) and the function is of the form (g(x)=f(x + 2)).

Step2: Analyze the options

  • Option A: Since (h=2>0) in (g(x)=f(x + 2)), the graph of (f(x)=x^{2}) is shifted 2 units to the left.
  • Option C: A vertical shift is of the form (y=f(x)+k) ((k\neq0)). For (g(x)=(x + 2)^{2}), it is not a vertical shift.
  • Option D: A shift to the right is of the form (y = f(x - h)) ((h>0)). But (g(x)=(x + 2)^{2}=f(x + 2)) is not of this form.

Answer:

The graph of (f(x)=x^{2}) should be shifted 2 units to the left.