Use transformations of $$ f(x)=x^{2} $$ to graph the following function. $$ g(x)=x^{2}-4 $$

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To graph the function $$ g(x) = x^2 - 4 $$ using transformations of the base function $$ f(x) = x^2 $$, we can follow these steps:

1. **Identify the base function**: The base function is $$ f(x) = x^2 $$, which is a parabola that opens upwards with its vertex at the origin (0, 0).

2. **Determine the transformation**: The function $$ g(x) = x^2 - 4 $$ can be seen as a vertical shift of the base function. Specifically, the "-4" indicates that the entire graph of $$ f(x) $$ is shifted downward by 4 units.

3. **Graph the transformations**:
   - Start with the graph of $$ f(x) = x^2 $$.
   - Identify the vertex of $$ f(x) $$, which is at (0, 0).
   - Shift the vertex down by 4 units. The new vertex of $$ g(x) $$ will be at (0, -4).

4. **Plot additional points**: To accurately graph $$ g(x) $$, we can calculate a few additional points:
   - For $$ x = -2 $$: 
     $$
     g(-2) = (-2)^2 - 4 = 4 - 4 = 0 \quad \text{(point: (-2, 0))}
     $$
   - For $$ x = -1 $$: 
     $$
     g(-1) = (-1)^2 - 4 = 1 - 4 = -3 \quad \text{(point: (-1, -3))}
     $$
   - For $$ x = 1 $$: 
     $$
     g(1) = (1)^2 - 4 = 1 - 4 = -3 \quad \text{(point: (1, -3))}
     $$
   - For $$ x = 2 $$: 
     $$
     g(2) = (2)^2 - 4 = 4 - 4 = 0 \quad \text{(point: (2, 0))}
     $$

5. **Sketch the graph**: 
   - Plot the vertex at (0, -4).
   - Plot the points (-2, 0), (-1, -3), (1, -3), and (2, 0).
   - Draw a smooth curve through these points to form the parabola.

The final graph of $$ g(x) = x^2 - 4 $$ is a parabola that opens upwards with its vertex at (0, -4) and intersects the x-axis at (-2, 0) and (2, 0).

This transformation illustrates how the graph of a function can be altered by shifting it vertically.
To graph $$ g(x) = x^2 - 4 $$, start with the base parabola $$ f(x) = x^2 $$ and shift it down by 4 units. The vertex moves from (0,0) to (0,-4). Plot additional points like (-2,0), (-1,-3), (1,-3), and (2,0), then draw a smooth upward-opening parabola through these points.

Use transformations of \( f(x)=x^{2} \) to graph the following function. \[ g(x)=x^{2}-4 \]

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