Problem 20 - Algebra 2

Question

Identify the focus and the directrix of the graph of each equation. $$ y=\frac{1}{2} x^{2} $$

Step-by-Step Solution

Answer

The focus of the parabola defined by the equation \(y=\frac{1}{2}x^{2}\) is at (0,2), and the directrix is the line \(y=-2\).

Step 1: Identify the type and direction of the parabola

In the given equation \(y=\frac{1}{2}x^{2}\), the square term is on x. Therefore, this is a vertical parabola. The positive coefficient of \(x^{2}\) indicates that the parabola opens upwards.

Step 2: Identify the vertex of the parabola

The equation does not contain any shifting (h,k), so the vertex of the parabola is the origin \((0,0)\).

Step 3: Calculate the focus of the parabola

The value of 'a' in the equation is \(a = \frac{1}{2}\). The coordinates of the focus based on the vertex \((h,k)\) and the value of \(a\) in a vertical parabola that opens upwards are \((h,k+\frac{1}{4a})\). So the focus of the parabola is \((0,0+\frac{1}{4*\frac{1}{2}})=(0,2)\). So the focus of the given parabola is at (0,2).

Step 4: Calculate the directrix of the parabola

For a parabola opening upwards or downwards, the equation of the directrix is \((y=k-\frac{1}{4a})\). Substituting the values, we get the equation of the directrix to be \((y=0-\frac{1}{4*\frac{1}{2}})\) or \(y=-2\).

Key Concepts

Vertex

The vertex of a parabola is a crucial point because it serves as the "tip" or "turning point" of the parabola. In simple terms, it's the point where the parabola changes direction. For vertical parabolas, which open either up or down, the vertex determines the lowest or highest point respectively.

In the equation given, \( y = \frac{1}{2} x^2 \), there is no addition or subtraction term attached to \( x^2 \). This means there is no horizontal or vertical shift of the parabola.

• Hence, the vertex is simply at the origin \((0,0)\).

The vertex also acts as a symmetrical point, meaning the parabola is mirrored across the line \( x = 0 \). Understanding the vertex is key to graphing any parabola effectively.

Focus

The focus of a parabola is one of its defining features. It is a point from which distances are measured to construct the parabola. For vertical parabolas, the focus lies along the axis of symmetry.

Given the equation \( y = \frac{1}{2} x^2 \), we identify the value of \( a \) which is \( \frac{1}{2} \) in this instance. The formula to find the focus of a vertical parabola opening upwards is given by:

• \((h, k + \frac{1}{4a})\)

In this case, replacing \( h \) and \( k \) with 0, the focus is calculated as:

• \(( 0, 0+ \frac{1}{4 \times \frac{1}{2}} ) = (0, 2)\)

The focus helps in understanding how the parabola is shaped and where it is pointed relative to its vertex.

Directrix

The directrix is a straight line that is part of the geometric definition of a parabola. For vertical parabolas, this line is horizontal and sits opposite to the direction in which the parabola opens.

For the given parabola \( y = \frac{1}{2} x^2 \), the directrix can be found using the formula:

• \( y = k - \frac{1}{4a} \)

Substituting the known values:

• \( y = 0 - \frac{1}{4 \times \frac{1}{2}} = -2 \)

Therefore, the equation of the directrix is \( y = -2 \). The directrix, in essence, works with the focus to determine the shape of the parabola, maintaining a constant distance according to the parabola's reflective property.

Vertical Parabola

A vertical parabola is one that opens either upwards or downwards, depending on the sign of the \( a \) term in the equation \( y = ax^2 \). With a positive \( a \), like in the equation \( y = \frac{1}{2} x^2 \), the parabola opens upwards.

Key characteristics of a vertical parabola include:

• The vertex as the lowest or highest point.
• The axis of symmetry, a vertical line through the vertex.
• The focus sitting along this axis of symmetry.
• The directrix being a horizontal line perpendicular to the axis of symmetry.

Vertical parabolas are commonly seen in physics (e.g., projectile motion) and engineering (e.g., satellite dishes) because of their unique reflective properties, directing signals or forces towards the focus.