The graph below displays a quadratic function. Identify the equation of the functi and determine the following features of the function: intercepts, vertex, and end behavior.

I'd be happy to help you analyze the quadratic function! However, since I can't see the graph you're referring to, I'll guide you through the general process of identifying the equation of a quadratic function and determining its key features. If you provide specific details from your graph, I can give a more tailored response. ### General Form of a Quadratic Function A quadratic function typically has the form: \[ f(x) = ax^2 + bx + c \] Alternatively, it can be expressed in **vertex form**: \[ f(x) = a(x - h)^2 + k \] where: - \((h, k)\) is the **vertex** of the parabola. - \(a\) determines the **opening direction** and **width** of the parabola. ### Steps to Identify the Equation and Features 1. **Determine the Vertex:** - Locate the highest or lowest point on the graph. This point is the vertex \((h, k)\). 2. **Find the Y-Intercept:** - The y-intercept occurs where \(x = 0\). This is the point \((0, c)\). 3. **Find the X-Intercepts (Roots):** - These are the points where the graph crosses the x-axis. Solve \(f(x) = 0\) to find these values. 4. **Determine the Orientation:** - If \(a > 0\), the parabola opens **upwards**. - If \(a < 0\), the parabola opens **downwards**. 5. **Find Additional Points (if needed):** - Use other points on the graph to set up equations and solve for unknown coefficients if the vertex form isn't directly applicable. ### Determining the Equation **Using Vertex Form:** If you know the vertex \((h, k)\) and another point \((x, y)\) on the graph: 1. Start with the vertex form: \[ f(x) = a(x - h)^2 + k \] 2. Plug in the known point to solve for \(a\): \[ y = a(x - h)^2 + k \] 3. Once \(a\) is known, you have the complete equation. **Conversion to Standard Form:** If needed, expand the vertex form to convert it to the standard form \(ax^2 + bx + c\). ### Determining Key Features - **Intercepts:** - **Y-Intercept:** Directly read from the graph where \(x = 0\). - **X-Intercepts:** Solve \(f(x) = 0\) using the quadratic formula, factoring, or completing the square. - **Vertex:** - Located at \((h, k)\) in the vertex form. - **End Behavior:** - If \(a > 0\): As \(x \to \pm\infty\), \(f(x) \to +\infty\). - If \(a < 0\): As \(x \to \pm\infty\), \(f(x) \to -\infty\). ### Example **Given:** - Vertex at \((2, -3)\) - Y-intercept at \((0, 1)\) - Parabola opens upwards. **Steps:** 1. **Use Vertex Form:** \[ f(x) = a(x - 2)^2 - 3 \] 2. **Use Y-Intercept to Find \(a\):** \[ 1 = a(0 - 2)^2 - 3 \] \[ 1 = 4a - 3 \] \[ 4a = 4 \] \[ a = 1 \] 3. **Equation:** \[ f(x) = (x - 2)^2 - 3 \] 4. **Convert to Standard Form (optional):** \[ f(x) = x^2 - 4x + 4 - 3 \] \[ f(x) = x^2 - 4x + 1 \] 5. **Find X-Intercepts:** \[ x^2 - 4x + 1 = 0 \] \[ x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \] 6. **End Behavior:** - Since \(a = 1 > 0\), as \(x \to \pm\infty\), \(f(x) \to +\infty\). ### Conclusion To provide a specific equation and accurately determine the features of your quadratic function, please share key details from the graph such as: - Coordinates of the vertex - Y-intercept - X-intercepts (if available) - Any other points that lie on the graph With this information, I can help you find the exact equation and analyze its characteristics comprehensively.