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

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To graph the function $$ g(x) = x^{2} - 2 $$ using transformations of the parent function $$ f(x) = x^{2} $$, follow these steps:

### 1. Start with the Parent Function: $$ f(x) = x^{2} $$

- **Graph Characteristics:**
  - **Shape:** A parabola opening upwards.
  - **Vertex:** At the origin $$(0, 0)$$.
  - **Axis of Symmetry:** The y-axis ($$x = 0$$).
  - **Intercepts:** 
    - **Y-intercept:** $$(0, 0)$$.
    - **X-intercepts:** $$(0, 0)$$ (since it's only touching the origin).

### 2. Apply the Transformation: Vertical Shift Downward by 2 Units

The function $$ g(x) = x^{2} - 2 $$ represents a vertical shift of the parent function downward by 2 units. Here's how this transformation affects the graph:

- **New Vertex:**
  - The vertex of $$ f(x) = x^{2} $$ is at $$(0, 0)$$.
  - After shifting down by 2 units, the new vertex of $$ g(x) $$ is at $$(0, -2)$$.

- **Axis of Symmetry:**
  - Remains the same as the parent function, which is the y-axis ($$x = 0$$).

- **Intercepts:**
  - **Y-intercept:**
    - Set $$ x = 0 $$ in $$ g(x) $$:
    $$
    g(0) = 0^{2} - 2 = -2
    $$
    - So, the y-intercept is at $$(0, -2)$$.
  
  - **X-intercepts:**
    - Set $$ g(x) = 0 $$:
    $$
    x^{2} - 2 = 0 \implies x^{2} = 2 \implies x = \pm \sqrt{2}
    $$
    - So, the x-intercepts are at $$(\sqrt{2}, 0)$$ and $$(-\sqrt{2}, 0)$$.

### 3. Plotting Key Points and Sketching the Graph

- **Vertex:** $$(0, -2)$$
- **Y-intercept:** $$(0, -2)$$
- **X-intercepts:** $$(\sqrt{2}, 0)$$ and $$(-\sqrt{2}, 0)$$
- **Additional Points:**
  - Choose a few more x-values to get corresponding y-values for precision.
    - $$ x = 1 $$:
      $$
      g(1) = 1^{2} - 2 = -1 \quad \Rightarrow \quad (1, -1)
      $$
    - $$ x = -1 $$:
      $$
      g(-1) = (-1)^{2} - 2 = -1 \quad \Rightarrow \quad (-1, -1)
      $$
    - $$ x = 2 $$:
      $$
      g(2) = 2^{2} - 2 = 2 \quad \Rightarrow \quad (2, 2)
      $$
    - $$ x = -2 $$:
      $$
      g(-2) = (-2)^{2} - 2 = 2 \quad \Rightarrow \quad (-2, 2)
      $$

- **Sketching the Graph:**
  1. Plot the vertex at $$(0, -2)$$.
  2. Plot the x-intercepts at $$(\sqrt{2}, 0)$$ and $$(-\sqrt{2}, 0)$$.
  3. Plot the additional points: $$(1, -1)$$, $$(-1, -1)$$, $$(2, 2)$$, and $$(-2, 2)$$.
  4. Draw a smooth, symmetric parabola opening upwards through these points.

### 4. Summary of Transformations

- **Parent Function:** $$ f(x) = x^{2} $$
- **Transformation Applied:** 
  - **Vertical Shift:** Downward by 2 units.
- **Transformed Function:** $$ g(x) = x^{2} - 2 $$

### 5. Visual Representation

While I can't provide a visual graph here, following the steps above with graphing paper or a graphing tool will help you accurately plot $$ g(x) = x^{2} - 2 $$. The key takeaway is that subtracting 2 from $$ x^{2} $$ shifts the entire parabola downward by 2 units without altering its shape or orientation.

### 6. Example Graph Description

Imagine the standard parabola $$ f(x) = x^{2} $$ centered at the origin. When you apply the transformation to get $$ g(x) = x^{2} - 2 $$, the entire parabola moves straight down so that the new lowest point (vertex) is at $$(0, -2)$$. The arms of the parabola remain equally wide and continue to open upwards, intersecting the x-axis at $$ x = \sqrt{2} $$ and $$ x = -\sqrt{2} $$.

### 7. Final Graph Characteristics for $$ g(x) = x^{2} - 2 $$

- **Vertex:** $$(0, -2)$$
- **Axis of Symmetry:** $$ x = 0 $$ (y-axis)
- **Y-intercept:** $$(0, -2)$$
- **X-intercepts:** $$(\sqrt{2}, 0)$$ and $$(-\sqrt{2}, 0)$$
- **Direction:** Opens upwards
- **Width:** Same as the parent function $$ f(x) = x^{2} $$

By understanding and applying these transformations, you can graph $$ g(x) = x^{2} - 2 $$ accurately and recognize how it relates to the parent function $$ f(x) = x^{2} $$.
To graph $$ g(x) = x^{2} - 2 $$, start with the parent function $$ f(x) = x^{2} $$ and shift it downward by 2 units. This moves the vertex from $$(0, 0)$$ to $$(0, -2)$$, keeping the parabola's shape and symmetry the same. The new graph will have a y-intercept at $$(0, -2)$$ and x-intercepts at $$(\sqrt{2}, 0)$$ and $$(-\sqrt{2}, 0)$$.

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

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