Let's determine which graph corresponds to the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex] by breaking it down step by step.
### Step 1: Identify the Parent Function
The parent function here is [tex]\( f(x) = x^2 \)[/tex], which is a parabola with its vertex at the origin (0, 0) and which opens upwards.
### Step 2: Horizontal Shift
In the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex], the term [tex]\( (x+2) \)[/tex] means that the graph of the parent function [tex]\( x^2 \)[/tex] is shifted horizontally. Specifically, the [tex]\(+2\)[/tex] inside the parentheses indicates a shift to the left by 2 units. Therefore, the vertex of the parabola moves from (0, 0) to (-2, 0).
### Step 3: Vertical Shift
The [tex]\( -3 \)[/tex] outside the squared term indicates a vertical shift. This means that the entire graph is shifted downward by 3 units. Therefore, the vertex of the parabola moves from (-2, 0) to (-2, -3).
### Step 4: Graph Characteristics
From the above steps, we know that:
- The vertex of the parabola is at (-2, -3).
- The parabola opens upwards (since the coefficient of the squared term is positive).
### Summary
To summarize, the graph of [tex]\( f(x) = (x+2)^2 - 3 \)[/tex]:
- Has a vertex at (-2, -3).
- Is an upwards-opening parabola.
These steps should help you identify which graph corresponds to the function [tex]\( f(x) = (x+2)^2 - 3 \)[/tex]. Look for a parabola with a vertex at (-2, -3) and that opens upwards.
