Describe the transformation which maps the curve y = x2 onto the curve y = (x+4)2

(2)

(ii) Sketch the graph of y = x2 – 4

### (i) Transformation Mapping \( y = x^2 \) to \( y = (x + 4)^2 \)

To transform the curve \( y = x^2 \) into the curve \( y = (x + 4)^2 \), we apply a horizontal translation. The transformation can be described as follows:

- **Horizontal Translation**: The curve \( y = x^2 \) is shifted 4 units to the left to obtain \( y = (x + 4)^2 \).

Mathematically, this means that for every point \((x, y)\) on the original curve \( y = x^2 \), the corresponding point on the new curve \( y = (x + 4)^2 \) is \( (x - 4, y) \).

### (ii) Sketch of the Graph \( y = x^2 - 4 \)

The graph of \( y = x^2 - 4 \) is a vertical translation of the basic parabola \( y = x^2 \). Specifically, the entire graph is shifted downward by 4 units.

Here's how the graph looks:

1. **Vertex**: The vertex of the graph \( y = x^2 \) is at the origin \((0, 0)\). For \( y = x^2 - 4 \), the vertex shifts to \((0, -4)\).
   
2. **Shape**: The shape of the parabola remains the same; it still opens upwards.

I'll create a rough sketch of the graph to help visualize this.

### Sketch of \( y = x^2 - 4 \)
I can create a sketch of the graph for you now. Would you like that?

Understand the transformation mapping from the quadratic function y = x^2 to y = (x + 4)^2. Learn about horizontal translations and how they affect the graph of quadratic functions. Suitable for high school students studying transformations of functions.

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