Graph the parabola and the axis of symmetry. Use a the axis of symmetry. Use the graph to write the dor \( f(x)=-2 x^{2}+5 \) Part: \( 0 / 5 \)

To graph the parabola \( f(x) = -2x^{2} + 5 \) and identify its **axis of symmetry**, follow these steps: ### 1. **Understanding the Parabola** The given quadratic function is in the form: \[ f(x) = ax^{2} + bx + c \] where: - \( a = -2 \) - \( b = 0 \) (since there's no \( x \) term) - \( c = 5 \) Since \( a = -2 \) is negative, the parabola opens **downwards**. ### 2. **Finding the Vertex** The vertex of a parabola \( f(x) = ax^{2} + bx + c \) is given by: \[ \text{Vertex } (h, k) \] where: \[ h = -\frac{b}{2a} \] \[ k = f(h) \] For our function: \[ h = -\frac{0}{2(-2)} = 0 \] \[ k = f(0) = -2(0)^2 + 5 = 5 \] **Vertex:** \( (0, 5) \) ### 3. **Determining the Axis of Symmetry** The **axis of symmetry** of a parabola is a vertical line that passes through the vertex. Its equation is: \[ x = h \] For our vertex \( (0, 5) \): \[ \text{Axis of Symmetry: } x = 0 \] ### 4. **Plotting the Parabola** 1. **Vertex:** Plot the vertex at \( (0, 5) \). 2. **Axis of Symmetry:** Draw a dashed vertical line through the vertex along \( x = 0 \). 3. **Additional Points:** - Choose values for \( x \) on either side of the axis of symmetry to find corresponding \( y \)-values. For example: | \( x \) | \( f(x) = -2x^{2} + 5 \) | |-------|--------------------------| | -2 | \( -2(-2)^2 + 5 = -8 + 5 = -3 \) | | -1 | \( -2(-1)^2 + 5 = -2 + 5 = 3 \) | | 0 | \( 5 \) (vertex) | | 1 | \( -2(1)^2 + 5 = -2 + 5 = 3 \) | | 2 | \( -2(2)^2 + 5 = -8 + 5 = -3 \) | 4. **Symmetry:** Plot these points on both sides of the axis of symmetry to maintain symmetry. 5. **Shape:** Connect the points with a smooth, U-shaped curve opening downwards. ### 5. **Graph Illustration** While I can't provide a visual graph here, here's a description of how it would look: - **Vertex:** Highest point at (0, 5). - **Axis of Symmetry:** A vertical dashed line along \( x = 0 \). - **Points:** - Left of the vertex: (-2, -3) and (-1, 3). - Right of the vertex: (1, 3) and (2, -3). - **Parabola Shape:** Downward-facing "U" passing through these points. ### 6. **Summary** - **Axis of Symmetry:** \( x = 0 \) - **Vertex:** \( (0, 5) \) - **Direction:** Opens downward. ### 7. **Final Answer** After graphing \( f(x) = -2x^{2} + 5 \), the **axis of symmetry** is the vertical line: \[ \boxed{x = 0} \]