Move the point and the vertex to graph the function $$ f(x)=2(x+1)^{2}+2 $$

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To graph the function $$ f(x) = 2(x + 1)^2 + 2 $$, we can follow these steps:

1. **Identify the vertex**: The function is in vertex form, which is $$ f(x) = a(x - h)^2 + k $$, where $$ (h, k) $$ is the vertex of the parabola. In this case, we can see that:
   - $$ a = 2 $$
   - $$ h = -1 $$
   - $$ k = 2 $$

Therefore, the vertex of the function is at the point $$ (-1, 2) $$.

2. **Determine the direction of the parabola**: Since $$ a = 2 $$ is positive, the parabola opens upwards.

3. **Find additional points**: To graph the function accurately, we can find a few additional points by substituting values of $$ x $$ into the function.

Let's calculate $$ f(x) $$ for $$ x = -2, -1, 0, 1 $$:

- For $$ x = -2 $$:
     $$
     f(-2) = 2(-2 + 1)^2 + 2 = 2(-1)^2 + 2 = 2 \cdot 1 + 2 = 4
     $$
   - For $$ x = -1 $$:
     $$
     f(-1) = 2(-1 + 1)^2 + 2 = 2(0)^2 + 2 = 2
     $$
   - For $$ x = 0 $$:
     $$
     f(0) = 2(0 + 1)^2 + 2 = 2(1)^2 + 2 = 2 \cdot 1 + 2 = 4
     $$
   - For $$ x = 1 $$:
     $$
     f(1) = 2(1 + 1)^2 + 2 = 2(2)^2 + 2 = 2 \cdot 4 + 2 = 8 + 2 = 10
     $$

4. **Plot the points**: Now we have the following points to plot:
   - Vertex: $$ (-1, 2) $$
   - Point 1: $$ (-2, 4) $$
   - Point 2: $$ (0, 4) $$
   - Point 3: $$ (1, 10) $$

5. **Draw the graph**: Connect the points with a smooth curve to form the parabola.

Now, let's summarize the points we found:
- Vertex: $$ (-1, 2) $$
- Additional points: $$ (-2, 4) $$, $$ (0, 4) $$, $$ (1, 10) $$

The graph of the function $$ f(x) = 2(x + 1)^2 + 2 $$ is a parabola that opens upwards with its vertex at $$ (-1, 2) $$.
To graph the function $$ f(x) = 2(x + 1)^2 + 2 $$, follow these steps: 1. **Identify the vertex**: The vertex is at $$ (-1, 2) $$. 2. **Plot the vertex**. 3. **Find additional points**: - $$ (-2, 4) $$ - $$ (0, 4) $$ - $$ (1, 10) $$ 4. **Draw the parabola** by connecting these points with a smooth curve. The graph is an upward-opening parabola with its vertex at $$ (-1, 2) $$.

Move the point and the vertex to graph the function \( f(x)=2(x+1)^{2}+2 \)

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