ter 1 $$ 1 \leftarrow $$ 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 Shape:** The graph of $$ f(x) = x^{2} $$ is a parabola that opens upward.
- **Vertex:** The vertex is at the origin $$(0, 0)$$.
- **Key Points:** Some key points on this graph include $$(-2, 4)$$, $$(-1, 1)$$, $$(1, 1)$$, and $$(2, 4)$$.

### 2. **Identify the Transformation**
- The function $$ g(x) = x^{2} + 2 $$ involves **vertical translation**.
- **Translation Rule:** Adding a constant outside the function (in this case, $$ +2 $$) shifts the graph **vertically**.
  - If the constant is positive ($$+2$$), the graph moves **upward**.
  - If the constant is negative ($$-2$$), the graph would move **downward**.

### 3. **Apply the Transformation**
- **Shift Upward by 2 Units:**
  - Since we are adding 2, move every point of the parent parabola $$ f(x) = x^{2} $$ **up** by 2 units.
  - **New Vertex:** The vertex moves from $$(0, 0)$$ to $$(0, 2)$$.
  - **Transformed Key Points:**
    - $$(-2, 4) \rightarrow (-2, 6)$$
    - $$(-1, 1) \rightarrow (-1, 3)$$
    - $$(1, 1) \rightarrow (1, 3)$$
    - $$(2, 4) \rightarrow (2, 6)$$

### 4. **Plot the Transformed Points and Draw the Parabola**
- Plot the new vertex $$(0, 2)$$ and the transformed key points.
- Draw a smooth curve through these points to form the new parabola.
- **Shape:** The shape remains the same (still opens upward), just shifted up.

### 5. **Final Graph of $$ g(x) = x^{2} + 2 $$**
- The graph is a parabola identical in shape to $$ f(x) = x^{2} $$ but located 2 units above it.
  
#### **Graph Illustration:**

```
y
|
6 |        *          
5 |                    
4 |      *   *
3 |    *       *
2 |  *-----------*  <-- Vertex at (0,2)
1 |  
0 |------------------------ x
   -2  -1   0   1   2
```

- **Points Plotted:**
  - $$(-2, 6)$$, $$(-1, 3)$$, $$(0, 2)$$, $$(1, 3)$$, $$(2, 6)$$

### **Summary of Transformation:**
- **Type of Transformation:** Vertical shift.
- **Direction:** Upward.
- **Magnitude:** 2 units.

By understanding and applying these transformations, you can graph $$ g(x) = x^{2} + 2 $$ effectively using 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 upward by 2 units. This moves the vertex from $$(0,0)$$ to $$(0,2)$$ and raises all points on the parabola by 2 units. The resulting graph is a parabola opening upward with its vertex at $$(0,2)$$ and key points at $$(-2,6)$$, $$(-1,3)$$, $$(1,3)$$, and $$(2,6)$$.

ter 1 \( 1 \leftarrow \) Use transformations of \( f(x)=x^{2} \) to graph the following function. \( g(x)=x^{2}+2 \)

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