4. Graph the parabola. Include the vertex and two accurate points on each side of the axis of symmetry. \( f(x)=2(x-3)^{2}-6 \)

To graph the parabola defined by the function \( f(x) = 2(x - 3)^2 - 6 \), we'll follow these steps: 1. **Identify the Vertex** 2. **Determine the Axis of Symmetry** 3. **Calculate Additional Points** 4. **Plot the Points and Draw the Parabola** Let's go through each step in detail. --- ### 1. Identify the Vertex The given quadratic function is in **vertex form**, which is: \[ f(x) = a(x - h)^2 + k \] Here, \((h, k)\) represents the vertex of the parabola. Comparing with the given function: \[ f(x) = 2(x - 3)^2 - 6 \] - \( h = 3 \) - \( k = -6 \) **Vertex:** \( (3, -6) \) --- ### 2. Determine the Axis of Symmetry The **axis of symmetry** is a vertical line that passes through the vertex. Its equation is: \[ x = h \] For our function: \[ x = 3 \] This line divides the parabola into two mirror-image halves. --- ### 3. Calculate Additional Points To accurately graph the parabola, we'll find two points on each side of the axis of symmetry. #### Selecting x-values: Choose x-values around the vertex \( x = 3 \): - **Left of Vertex:** \( x = 1 \) and \( x = 2 \) - **Right of Vertex:** \( x = 4 \) and \( x = 5 \) #### Calculating f(x) for Each x-value: 1. **For \( x = 1 \):** \[ f(1) = 2(1 - 3)^2 - 6 = 2(4) - 6 = 8 - 6 = 2 \] **Point:** \( (1, 2) \) 2. **For \( x = 2 \):** \[ f(2) = 2(2 - 3)^2 - 6 = 2(1) - 6 = 2 - 6 = -4 \] **Point:** \( (2, -4) \) 3. **For \( x = 4 \):** \[ f(4) = 2(4 - 3)^2 - 6 = 2(1) - 6 = 2 - 6 = -4 \] **Point:** \( (4, -4) \) 4. **For \( x = 5 \):** \[ f(5) = 2(5 - 3)^2 - 6 = 2(4) - 6 = 8 - 6 = 2 \] **Point:** \( (5, 2) \) **Additional Points:** - Left Side: - \( (1, 2) \) - \( (2, -4) \) - Right Side: - \( (4, -4) \) - \( (5, 2) \) --- ### 4. Plot the Points and Draw the Parabola Here's a table of the key points to plot: | \( x \) | \( f(x) \) | Point | |---------|------------|------------| | 1 | 2 | (1, 2) | | 2 | -4 | (2, -4) | | 3 | -6 | (3, -6) **Vertex** | | 4 | -4 | (4, -4) | | 5 | 2 | (5, 2) | **Steps to Graph:** 1. **Draw the Coordinate Plane:** - Horizontal axis (x-axis) and vertical axis (y-axis). 2. **Plot the Vertex:** - Mark the point \( (3, -6) \). 3. **Draw the Axis of Symmetry:** - Draw a dashed vertical line through \( x = 3 \). 4. **Plot the Additional Points:** - Plot \( (1, 2) \), \( (2, -4) \), \( (4, -4) \), and \( (5, 2) \). 5. **Draw the Parabola:** - Connect the points smoothly, ensuring the parabola opens **upward** since the coefficient \( a = 2 \) is positive. - The graph should be symmetric about the axis \( x = 3 \). --- ### Visual Representation While I can't provide a visual graph here, following the steps above with graph paper or a graphing tool will yield the correct parabola. Here's a rough sketch of how the points align: ``` y | 2 | * * 1 | 0 | -2| -4 | * * -6 | V -8| ------------------------- x 1 2 3 4 5 ``` - **\( V \)** represents the Vertex at \( (3, -6) \). - Stars (\(*\)) represent the additional points. --- ### Summary of Key Features - **Vertex:** \( (3, -6) \) - **Axis of Symmetry:** \( x = 3 \) - **Additional Points:** - Left: \( (1, 2) \) and \( (2, -4) \) - Right: \( (4, -4) \) and \( (5, 2) \) - **Direction:** Opens **upward** (since \( a = 2 > 0 \)) By plotting these points and connecting them with a smooth, U-shaped curve, you'll accurately graph the parabola defined by \( f(x) = 2(x - 3)^2 - 6 \).