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Part 1
Use the graph of y equals x squared to graph the quadratic function.
f left parenthesis x right parenthesis equals left parenthesis x plus 6 right parenthesis squared plus 2
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Part 1
Use the graphing tool to graph the function.

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To graph the quadratic function \( f(x) = (x + 6)^2 + 2 \), you'll follow these steps based on the transformation of the basic quadratic function \( y = x^2 \):

### 1. **Identify the Transformations:**
   - **Horizontal Shift:** The \( x + 6 \) inside the squared term indicates a horizontal shift. Specifically, the graph of \( y = x^2 \) will shift 6 units to the **left**.
   - **Vertical Shift:** The \( +2 \) outside the squared term indicates a vertical shift. The graph will move 2 units **up**.

### 2. **Determine the Vertex:**
   - The vertex of the basic function \( y = x^2 \) is at the origin \((0, 0)\).
   - After the transformations, the vertex of \( f(x) = (x + 6)^2 + 2 \) will be shifted to the point \((-6, 2)\).

### 3. **Plot the Vertex:**
   - Start by plotting the vertex at \((-6, 2)\).

### 4. **Plot Additional Points:**
   - Choose a few x-values around the vertex and calculate the corresponding y-values.
   - For example:
     - If \( x = -7 \): \( f(-7) = (-7 + 6)^2 + 2 = 1^2 + 2 = 3 \).
     - If \( x = -5 \): \( f(-5) = (-5 + 6)^2 + 2 = 1^2 + 2 = 3 \).
     - If \( x = -4 \): \( f(-4) = (-4 + 6)^2 + 2 = 2^2 + 2 = 6 \).

### 5. **Sketch the Parabola:**
   - Connect the points with a smooth curve, ensuring that it opens upwards, as the coefficient of \( x^2 \) is positive.

### 6. **Graph the Function:**
   - Use the graphing tool to plot the points and draw the curve.

**Graph Summary:**
- **Vertex:** \((-6, 2)\)
- **Direction:** Opens upwards.
- **Shifts:** Left 6 units, up 2 units.

Would you like more details or have any questions?

### Follow-up Questions:
1. How would the graph change if the equation was \( f(x) = (x - 6)^2 + 2 \)?
2. What is the axis of symmetry for the function \( f(x) = (x + 6)^2 + 2 \)?
3. How does the coefficient of the squared term affect the graph's shape?
4. What are the key differences between the graph of \( f(x) = x^2 \) and \( f(x) = (x + 6)^2 + 2 \)?
5. Can you derive the function's formula from a different vertex and transformation?

**Tip:** Always start by identifying the vertex and direction of the parabola when graphing quadratic functions.

Learn how to graph the quadratic function f(x) = (x + 6)^2 + 2 with step-by-step instructions. This problem involves understanding transformations of quadratic functions, including horizontal and vertical shifts. Suitable for high school students studying quadratic functions.

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