Graph each parabola. Include the vertex and at least three precise points on each side of the line of symmetry. Then describe the transformation in words. $$ g(x)=x^{2}-3 $$

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Certainly! Let's graph the parabola $$ g(x) = x^2 - 3 $$ by identifying its key features, plotting specific points, and describing its transformation.

## 1. Identify the Vertex

The standard form of a parabola is:
$$ y = a(x - h)^2 + k $$
where $$(h, k)$$ is the vertex.

For $$ g(x) = x^2 - 3 $$:
- $$ a = 1 $$ (which means the parabola opens upward)
- $$ h = 0 $$
- $$ k = -3 $$

**Vertex:** $$(0, -3)$$

## 2. Determine the Line of Symmetry

The line of symmetry is a vertical line that passes through the vertex.

**Line of Symmetry:** $$ x = 0 $$

## 3. Plot Points on Each Side of the Line of Symmetry

To get a clear picture of the parabola, we'll calculate and plot points on both sides of the line of symmetry ($$ x = 0 $$).

### Points to the Right of the Line of Symmetry ($$ x > 0 $$):

1. **When $$ x = 1 $$:**
   $$ g(1) = (1)^2 - 3 = 1 - 3 = -2 $$
   **Point:** $$(1, -2)$$

2. **When $$ x = 2 $$:**
   $$ g(2) = (2)^2 - 3 = 4 - 3 = 1 $$
   **Point:** $$(2, 1)$$

3. **When $$ x = 3 $$:**
   $$ g(3) = (3)^2 - 3 = 9 - 3 = 6 $$
   **Point:** $$(3, 6)$$

### Points to the Left of the Line of Symmetry ($$ x < 0 $$):

1. **When $$ x = -1 $$:**
   $$ g(-1) = (-1)^2 - 3 = 1 - 3 = -2 $$
   **Point:** $$(-1, -2)$$

2. **When $$ x = -2 $$:**
   $$ g(-2) = (-2)^2 - 3 = 4 - 3 = 1 $$
   **Point:** $$(-2, 1)$$

3. **When $$ x = -3 $$:**
   $$ g(-3) = (-3)^2 - 3 = 9 - 3 = 6 $$
   **Point:** $$(-3, 6)$$

## 4. Summary of Points

- **Vertex:** $$(0, -3)$$
- **Right Side Points:**
  - $$(1, -2)$$
  - $$(2, 1)$$
  - $$(3, 6)$$
- **Left Side Points:**
  - $$(-1, -2)$$
  - $$(-2, 1)$$
  - $$(-3, 6)$$

## 5. Graphing the Parabola

While I can't provide a visual graph here, you can plot the points listed above on a coordinate plane:

1. Plot the **vertex** at $$(0, -3)$$.
2. Plot the **right side points**: $$(1, -2)$$, $$(2, 1)$$, and $$(3, 6)$$.
3. Plot the **left side points**: $$(-1, -2)$$, $$(-2, 1)$$, and $$(-3, 6)$$.
4. Draw a smooth, U-shaped curve through these points, ensuring the parabola is symmetric about the line $$ x = 0 $$.

## 6. Describe the Transformation

The given parabola $$ g(x) = x^2 - 3 $$ is a transformation of the basic quadratic function $$ f(x) = x^2 $$. Here's how it has been transformed:

- **Vertical Shift Downward by 3 Units:** The entire graph of $$ f(x) = x^2 $$ has been moved **3 units downwards** to become $$ g(x) = x^2 - 3 $$. This shifts the vertex from $$(0, 0)$$ to $$(0, -3)$$.

There are **no horizontal shifts** or **reflections** involved since there are no coefficients affecting the $$ x $$ term or negative signs altering the direction of the parabola.

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**Visual Representation:**

If you plot the points and draw the parabola as described, it will resemble the standard upward-opening parabola shifted down by 3 units, centered along the y-axis.
The parabola $$ g(x) = x^2 - 3 $$ has its vertex at $$(0, -3)$$ and is symmetric about the y-axis. Points on the right side include $$(1, -2)$$, $$(2, 1)$$, and $$(3, 6)$$, while points on the left side include $$(-1, -2)$$, $$(-2, 1)$$, and $$(-3, 6)$$. The parabola is shifted downward by 3 units from the standard $$ y = x^2 $$ graph.

Graph each parabola. Include the vertex and at least three precise points on each side of the line of symmetry. Then describe the transformation in words. \[ g(x)=x^{2}-3 \]

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