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

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 \( (-1, 2) \).