which phrase best describes the translation from the graph y = (x + 2)^2 to the graph of y = x^2 + 3?\n2…

Explanation:

Step1: Analyze horizontal translation

For a quadratic function in vertex - form $y=a(x - h)^2+k$, the graph of $y=(x + 2)^2$ has vertex at $(-2,0)$ and the graph of $y=x^2$ has vertex at $(0,0)$. To go from $y=(x + 2)^2$ to $y=x^2$, we set $x+2$ to $x$. Solving $x+2=x_1$ for $x$ in terms of $x_1$ gives $x=x_1 - 2$. This means a shift of 2 units to the right.

Step2: Analyze vertical translation

The graph of $y=x^2$ is shifted to $y=x^2+3$. For a function $y = f(x)$ to $y=f(x)+c$ where $c>0$, the graph is shifted $c$ units up. Here $c = 3$, so it is a 3 - unit up shift.

Answer:

2 units right and 3 units up