Question

below is the graph of y = x^2. translate it to make it the graph of y = (x + 4)^2+3.

Explanation:

Step1: Analyze horizontal translation

For the function $y=(x + 4)^{2}+3$, compared to $y = x^{2}$, the value inside the parentheses is $x+4$. According to the rule of horizontal translation of functions $y = f(x + h)$ (where $h>0$ shifts the graph of $y = f(x)$ to the left by $h$ units), here $h = 4$, so it shifts 4 units to the left.

Step2: Analyze vertical translation

The + 3 outside the parentheses in $y=(x + 4)^{2}+3$ means that according to the rule of vertical translation of functions $y=f(x)+k$ (where $k>0$ shifts the graph of $y = f(x)$ up by $k$ units), here $k = 3$, so it shifts 3 units up.

Answer:

Translate the graph of $y = x^{2}$ 4 units to the left and 3 units up.