Question

the black graph is y = f(x). choose the equation for the red graph. y = f(x + 3) y + 3 = f(x) y - 3 = f(x) y = f(x - 3)

Explanation:

Step1: Recall function - translation rule

For a function $y = f(x)$, a horizontal shift of $h$ units to the left is given by $y=f(x + h)$ and a shift of $h$ units to the right is given by $y = f(x - h)$. A vertical shift of $k$ units up is given by $y=f(x)+k$ and a shift of $k$ units down is given by $y=f(x)-k$.

Step2: Analyze the graph shift

The vertex of the black - graph $y = f(x)$ is at $(0,0)$ and the vertex of the red - graph is at $(-3,0)$. This means the graph of $y = f(x)$ has been shifted 3 units to the left.
According to the horizontal - shift rule, if the graph of $y = f(x)$ is shifted 3 units to the left, the equation of the new graph is $y=f(x + 3)$.

Answer:

$y = f(x + 3)$