Problem 45 - College Algebra

Question

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$y^{2}-2 y+12 x-35=0$$

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x =\frac{1}{12}(y-1)^2 -3\). The vertex of the parabola is located at \((-3,1)\), the focus at \((-143/48, 1)\), and the directrix is the line \(x=-145/48\). After plotting these points and the line, you draw the parabola to complete the graph.

Step 1: Converting to standard form

To convert the given equation into standard form, we need to complete the square. The given equation is \(y^2-2y+12x-35=0\). Rearrange this equation to group y-terms together, then complete the square for y. This results in: \((y-1)^2=12x+36\). Now, rearrange the equation to set it equal to \(x\), which reveals the standard form of the equation of the parabola: \(12x=(y-1)^2-36\), \(x =\frac{1}{12}(y-1)^2 -3\).

Step 2: Identifying the vertex

The vertex of a parabola in the standard form \(x =\frac{1}{4p}(y-k)^2 +h\) is located at the point \((h,k)\). In this case, \(h = -3\) and \(k = 1\). Therefore, the vertex is at \((-3,1)\).

Step 3: Identifying the focus and directrix

In the standard form of our equation, \(4p = 1/12\). Thus, \(p = 1/48\). Because this is a positive number, and the parabola is in the standard form for a parabola that opens right or left, the parabola opens right when \(p\) is positive. The focus of the parabola is located at \((h+p, k)\), or \((-3+1/48, 1)=(-143/48, 1)\). The directrix of the parabola is the vertical line located at \(x=h-p\), or \(x=(-3-1/48)=(-145/48)\).

Step 4: Graphing the parabola

Draw a coordinate plane and mark the vertex of the parabola at (-3, 1). Illustrate the parabola opening to the right from the vertex. Locate and mark the focus, inside the parabola, at \((-143/48, 1)\). Draw the directrix, a vertical line, at \(x=(-145/48)\). Mark these elements clearly and label them. Check that the parabola is equidistant from the directrix and focus at all points to ensure accuracy.