Find the vertex, focus, and directrix of the following parabola. Graph the equation. Verify your graph using a graphing utility. \( (y+4)^{2}=4(x-1) \)

To analyze the given parabola \( (y+4)^{2}=4(x-1) \), we can rewrite it in the standard form of a parabola. The standard form for a parabola that opens to the right is: \[ (y - k)^2 = 4p(x - h) \] where \((h, k)\) is the vertex, \(p\) is the distance from the vertex to the focus (and also to the directrix), and the focus is located at \((h + p, k)\) while the directrix is the line \(x = h - p\). ### Step 1: Identify the vertex From the equation \( (y + 4)^2 = 4(x - 1) \), we can identify: - \(h = 1\) - \(k = -4\) Thus, the vertex of the parabola is: \[ (h, k) = (1, -4) \] ### Step 2: Identify \(p\) The coefficient \(4p\) in the equation is \(4\). Therefore, we have: \[ 4p = 4 \implies p = 1 \] ### Step 3: Find the focus The focus of the parabola is located at: \[ (h + p, k) = (1 + 1, -4) = (2, -4) \] ### Step 4: Find the directrix The directrix is given by the equation: \[ x = h - p = 1 - 1 = 0 \] ### Summary of the results - **Vertex**: \((1, -4)\) - **Focus**: \((2, -4)\) - **Directrix**: \(x = 0\) ### Step 5: Graph the parabola To graph the parabola, we can plot the vertex, focus, and directrix. The parabola opens to the right, and we can sketch it accordingly. 1. **Vertex**: Plot the point \((1, -4)\). 2. **Focus**: Plot the point \((2, -4)\). 3. **Directrix**: Draw a vertical line at \(x = 0\). ### Step 6: Verify using a graphing utility You can use a graphing utility (like Desmos, GeoGebra, or a graphing calculator) to plot the equation \( (y + 4)^2 = 4(x - 1) \) and verify that the vertex, focus, and directrix are as calculated. ### Final Graph The graph will show a parabola opening to the right with the vertex at \((1, -4)\), the focus at \((2, -4)\), and the directrix as the line \(x = 0\).