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

Explanation:

Step1: Recall 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. Here (f(x)=x^{2}) and (g(x)=(x + 2)^{2}=f(x + 2)) with (h = 2).

Step2: Identify key points of (f(x)=x^{2})

The vertex of (y=x^{2}) is ((0,0)). Other key points: when (x = 1,y=1); when (x=- 1,y = 1).

Step3: Apply the horizontal shift

For the vertex ((0,0)) of (y = x^{2}), after shifting (2) units to the left (using (x\to x+2)), the new vertex is ((-2,0)). For the point ((1,1)) of (y=x^{2}), let (x+2=1\Rightarrow x=-1), and (y = 1). For the point ((-1,1)) of (y=x^{2}), let (x + 2=-1\Rightarrow x=-3), and (y = 1).

Answer:

The graph of (g(x)=(x + 2)^{2}) is the graph of (f(x)=x^{2}) shifted (2) units to the left. The vertex is at ((-2,0)) and it has points ((-1,1)) and ((-3,1)) etc. (Use a graphing tool to plot these key - points and draw the parabola opening upwards).