Question: Describe the transformation of $f(x) = x^2$ represented by $g(x) = (x + 2)^2$.
The graph of $g$ is a translation 2 units left of the graph of $f$.
Identify the graph of each function.
top-left: $f$ is the blue parabola with vertex at $(0,0)$, and $g$ is the red parabola with vertex at $(2,0)$; this represents a shift 2 units right.
top-right: $f$ is the blue parabola with vertex at $(0,0)$, and $g$ is the red parabola with vertex at $(0,-2)$; this represents a shift 2 units down.
bottom-left: $f$ is the blue parabola with vertex at $(0,0)$, and $g$ is the red parabola with vertex at $(-2,0)$; this represents a shift 2 units left.
bottom-right: $f$ is the blue parabola with vertex at $(0,0)$, and $g$ is the red parabola with vertex at $(0,2)$; this represents a shift 2 units up.
1. **State the problem:** We want to describe the transformation of the function $f(x) = x^2$ when it changes to $g(x) = (x + 2)^2$ and identify which graph corresponds to this transformation.
2. **Recall the transformation rule:** For a function $f(x)$, the function $g(x) = f(x + h)$ represents a horizontal shift of the graph of $f$ by $-h$ units. That means if $h$ is positive, the graph shifts left by $h$ units; if $h$ is negative, it shifts right by $|h|$ units.
3. **Apply the rule:** Here, $g(x) = (x + 2)^2 = f(x + 2)$, so $h = 2$. This means the graph of $f$ shifts left by 2 units.
4. **Vertex analysis:** The vertex of $f(x) = x^2$ is at $(0,0)$.
The vertex of $g(x) = (x + 2)^2$ is at $(-2,0)$ because the inside of the square is zero when $x = -2$.
5. **Match with the given graphs:**
- Top-left: $g$ vertex at $(2,0)$ (shift right 2 units) — incorrect.
- Top-right: $g$ vertex at $(0,-2)$ (shift down 2 units) — incorrect.
- Bottom-left: $g$ vertex at $(-2,0)$ (shift left 2 units) — correct.
- Bottom-right: $g$ vertex at $(0,2)$ (shift up 2 units) — incorrect.
**Final answer:** The graph of $g(x) = (x + 2)^2$ is the bottom-left graph, representing a shift 2 units left of the graph of $f(x) = x^2$.