Problem 4 - Algebra for College Students

Question

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}=8 y $$

Step-by-Step Solution

Answer

Vertex: (0, 0), Focus: (0, 2), Directrix: y = -2.

Step 1: Identify the Parabola Type

The given equation is \(x^2 = 8y\). Recognize this as a vertical parabola because \(x\) is squared.

Step 2: Rewrite the Equation in Standard Form

For vertical parabolas, the standard form is \((x - h)^2 = 4p(y - k)\). Our equation is already in a comparable form: \(x^2 = 8y\). Notice \(8y\) can be written as \(4(2y)\), so \((h, k) = (0, 0)\) and \(4p = 8\), which gives \(p = 2\).

Step 3: Find the Vertex

The vertex of the parabola is given by the point \((h, k)\). Since \(h = 0\) and \(k = 0\) from the standard form comparison, the vertex is \((0, 0)\).

Step 4: Determine the Focus

For vertical parabolas, the focus is at the point \((h, k + p)\). Here, \(h = 0\), \(k = 0\), and \(p = 2\), so the focus is \((0, 2)\).

Step 5: Find the Directrix

The directrix of a vertical parabola is the line \(y = k - p\). Again, since \(k = 0\) and \(p = 2\), the directrix is \(y = -2\).

Step 6: Sketch the Graph

Plot the vertex at \((0, 0)\), the focus at \((0, 2)\), and draw the directrix line at \(y = -2\). Sketch the parabola opening upwards as it passes through the vertex and focuses according to the given equation.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction. It is essentially the "tip" of the parabola. For the given equation \(x^2 = 8y\), we start by rewriting it in its standard form. In a vertical parabola, this form is \((x - h)^2 = 4p(y - k)\). In our case, comparing \(x^2 = 8y\) to the standard form, it's clear that the vertex is at the origin point \((h, k) = (0, 0)\).
The vertex is crucial because it serves as the reference point for both the focus and the directrix, providing a balance between these components:

Focus of a Parabola

The focus of a parabola is a point located inside the curve from which the distances to any point on the parabola is equal to the distance from that point to a corresponding point on the directrix, a concept associated with reflective properties. Given our vertical parabola, the standard formula for the focus is \((h, k+p)\). We have already determined \(h = 0\) and \(k = 0\), and found \(p = 2\) by solving \(4p = 8\), leading us to the focus \((0, 2)\).
It's important to remember:

• The focus is always "inside" the parabola.
• For a vertical parabola, the focus moves vertically depending on \(p\).

Directrix of a Parabola

The directrix of a parabola is a straight line that is external to the parabola. It is used together with the focus to facilitate the curve being equidistant, thereby defining the parabolic shape. For vertical parabolas, the equation for the directrix is \(y = k - p\). With \(k = 0\) and \(p = 2\) as per our parabolic equation transformation, the directrix for this particular parabola becomes \(y = -2\).
This results in an equation for the directrix, which is a horizontal line for vertical parabolas:

• This line is perpendicular to the axis of symmetry of the parabola.
• It is always equidistant from the vertex as the focus is.