Problem 48 - Algebra and Trigonometry

Question

In Exercises 43–48, 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. $$ x^{2}+8 x-4 y+8=0 $$

Step-by-Step Solution

Answer

The parabola's equation in standard form is \(y = \frac{1}{4}(x+4)^2 - 2\). The vertex is at (-4, -2), the focus is at (-4, -1.75), and the equation of the directrix is \(y = -2.25\).

Step 1: Rearrange the given equation

Rearrange the given equation \(x^{2}+8 x-4 y+8=0\) to group the x-terms together: \(x^{2}+8 x = 4y - 8\)

Step 2: Complete the square on x

In order to complete the square on x, take half the coefficient of x, square it, and add it to both sides. Half of 8 is 4, and \(4^{2}\) is 16. Thus, \(x^{2}+8 x + 16 = 4y - 8 + 16\). This simplifies to \((x+4)^{2} = 4y + 8\)

Step 3: Rearrange into standard form

Rearrange the equation to isolate y on one side to put it in standard form: \((x+4)^{2} - 8 = 4y\), or \(y = \frac{(x+4)^2 - 8}{4}\) or equivalently, \(y = \frac{1}{4}(x+4)^2 - 2\)

Step 4: Identify the vertex, focus, and directrix

The vertex is given by (h, k), which in this case is (-4, -2). The number p (distance from vertex to focus or its negative to directrix) is equal to 1/4. So, the focus is (-4, -2 + 1/4) or (-4, -1.75); and the directrix is \(y = k - p = -2 - \frac{1}{4} = -2.25\)

Key Concepts

Completing the Square

Completing the square is a method used to simplify quadratic equations, allowing us to transform them into a more workable form. To complete the square, you identify the quadratic and linear terms, then manipulate them to form a perfect square trinomial. This involves taking the coefficient of the linear term (in this case, the x term) dividing it by 2, and squaring it.
For example, in the equation \(x^2 + 8x - 4y + 8 = 0\), our focus is on the terms involving \(x\): \(x^2 + 8x\). Here, the coefficient of \(x\) is 8.
Half of 8 is 4, and squaring it gives 16. We add and subtract 16 to form a complete square: \((x + 4)^2\). The right side of the equation is adjusted to maintain equality after completing the square.

Vertex of a Parabola

The vertex of a parabola is a crucial point that signifies the peak (either a minimum or maximum point) of the curve. In standard form \((x-h)^2 = 4p(y-k)\), the vertex is located at \((h, k)\). Identifying the vertex allows us to understand the symmetry and direction of the parabola.
For our equation, once we complete the square, it takes the form \((x+4)^2 = 4(y+2)\), meaning the vertex is at \((-4, -2)\). Knowing the vertex helps in sketching the graph because it serves as a reference point where the parabola changes direction.

Focus and Directrix of a Parabola

The focus and the directrix are properties of parabolas that define its shape and orientation. The focus is a point from which distances to any point on the parabola are measured. The directrix is a line equidistant from all points on the parabola that helps guide its curve.
For a parabola opening upwards or downwards, the relationship is given in the form \(y = a(x-h)^2 + k\) or equivalently \((x-h)^2 = 4p(y-k)\). Here, \(p\) represents the distance from the vertex to the focus as well as from the vertex to the directrix.
In the completed square equation \((x+4)^2 = 4(y+2)\), \(p\) is derived from \(4p = 4\), hence \(p = 1\). Thus, the focus is at \((-4,-1.75)\) and the directrix is the line \(y = -2.25\).

Standard Form of a Parabola

The standard form of a parabola is useful for quickly identifying its key characteristics like the vertex, direction, and orientation. For parabolas that open up or down, the standard form is \((x-h)^2 = 4p(y-k)\), where \((h,k)\) is the vertex and \(p\) determines the distance from the vertex to the focus and directrix.
After completing the square, our equation transforms into this standard form: \(y = \frac{1}{4}(x+4)^2 - 2\). This tells us the parabola opens upwards influenced by the positive coefficient of \(y\). It also makes it straightforward to identify the vertex \((-4, -2)\) and calculate \(p = 1\) as seen previously. Understanding this form is crucial for analyzing and graphing parabolas efficiently.