The problem asks you to graph the function $f(x) = (x + 2)^2 - 1$ using transformations starting from the graph of $y = x^2$, the basic parabola. Let's break it down step by step:

1. Base Function:

The base function is $y = x^2$, which is a standard parabola opening upwards with its vertex at the origin $(0, 0)$.

2. Transformation 1: Horizontal Shift

• The term $(x + 2)$ indicates a horizontal shift.
• Since the transformation involves $+2$, the graph shifts 2 units to the left.
• The new vertex is now at $(-2, 0)$.

3. Transformation 2: Vertical Shift

• The term $-1$ represents a vertical shift.
• This means the entire graph is shifted 1 unit down.
• The new vertex after this transformation is at $(-2, -1)$.

4. Summary of Transformations:

• Start with the graph of $y = x^2$ (a parabola).
• Shift the graph 2 units to the left.
• Then, shift the graph 1 unit downward.

The final function $f(x) = (x + 2)^2 - 1$ is a parabola with its vertex at $(-2, -1)$, opening upwards.

Would you like me to graph this function for you?

Related Questions:

1. How do vertical shifts affect the graph of a function?
2. What is the effect of horizontal shifts on a parabola?
3. How do you determine whether a transformation involves stretching or compressing?
4. Can you describe how a reflection would affect the function $y = x^2$?
5. How would the graph change if we had $f(x) = -(x + 2)^2 - 1$?

Tip:

Transformations on functions follow the order: first, handle horizontal changes (inside the parentheses), then vertical changes (outside).