Start from the original vertex. First adjust the $x$-coordinate for a 2-unit left shift, then adjust the $y$-coordinate for a 5-unit up shift.

Once you know the final vertex $(h, k)$ of $g$, plug those values into $y = (x - h)^2 + k$ and compare with the answer choices.

In Desmos, enter f(x) = (x - 3)^2. Notice the vertex at $(3, 0)$ by tapping or hovering over the lowest point of the graph.

Add a new expression like g(x) = (x - a)^2 + b. Desmos will create sliders for $a$ and $b$. This represents a parabola with vertex $(a, b)$.

Adjust the sliders so that the vertex of the new parabola is 2 units left and 5 units up from $(3, 0)$—that is, at the point $(1, 5)$. Once the vertex matches, read off the values of $a$ and $b$ from the sliders and write the corresponding equation. Choose the answer option whose equation is identical to that one.

The function $f(x) = (x - 3)^2$ is a parabola in vertex form.

In vertex form, $y = (x - h)^2 + k$ has its vertex at the point $(h, k)$.

So for $f(x) = (x - 3)^2$:

• $h = 3$
• $k = 0$

Therefore, the vertex of $f$ is at $(3, 0)$.

Translating a graph 2 units to the left moves every $x$-coordinate 2 less.

The original vertex is $(3, 0)$. After moving 2 units left:

• New $x$-coordinate: $3 - 2 = 1$
• $y$-coordinate stays $0$ for a purely horizontal move.

So after the left shift, the vertex becomes $(1, 0)$.

Translating a graph 5 units up moves every $y$-coordinate 5 more.

Starting from the intermediate vertex $(1, 0)$, move 5 units up:

• $x$-coordinate stays $1$
• New $y$-coordinate: $0 + 5 = 5$

So the final vertex of $g$ is $(1, 5)$.

A parabola that opens upward with vertex $(h, k)$ has equation

For $g$, the vertex is $(1, 5)$, so $h = 1$ and $k = 5$.

Substitute these values:

This matches answer choice D.