Problem 17 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The focus of the parabola y = x² is at (0, 1/4) and the equation of the directrix is y = -1/4.

Step 1: Identify the vertex

The given equation is 'y = x²', which is already in vertex form with h = 0 and k =0. Hence, the vertex of the parabola is at the origin (0,0).

Step 2: Identify the Focus

The focus of a parabola of the form y = a(x-h)² + k is given by the formula (h, k+1/4a). Here, a=1, h=0 and k=0, simplify the focus to (0,1/4).

Step 3: Identify the Directrix

The directrix of a parabola of the form y = a(x-h)² + k is a horizontal line given by the formula y = k - 1/4a. Here, a=1, h=0 and k=0. Hence, directrix is a line with equation: 'y = -1/4'.

Key Concepts

Vertex of a parabola

The vertex of a parabola is a key point and can be found in its equation. For the equation given as \( y = x^2 \), this is already in vertex form \( y = a(x-h)^2 + k \), where \( h \) and \( k \) are the coordinates of the vertex. In our case, both \( h \) and \( k \) equal zero, leading to the vertex being at point \( (0, 0) \).

This vertex represents the lowest point on the graph for this upward-opening parabola. It's important to note that the vertex here acts as the parabola's axis of symmetry, meaning the graph is a mirror image on either side of this point. Clearly identifying the vertex helps in graphing and better understanding the parabola's orientation and dimensions.

Focus of a parabola

The focus of a parabola is a point which, along with the directrix, helps in defining the shape and orientation of the parabola. For the standard form of a parabolic equation \( y = a(x-h)^2 + k \), the focus can be found using the formula \( (h, k+1/4a) \).

In our equation \( y = x^2 \), the values of \( h \) and \( k \) are \( 0 \), and \( a \) is \( 1 \). Substituting these values into the formula, the focus results in \( (0, 1/4) \).

This means the focus is located a quarter unit above the vertex. The focus is crucial because any point on the parabola is equidistant from the focus and the directrix, maintaining the parabola's definition. This property makes it essential in constructing parabolas accurately.

Directrix of a parabola

The directrix is a line that, alongside the focus, defines a parabola. It is a unique line that lies opposite the focus, and each point on the parabola is equidistant both from the focus and this line.

For the standard parabolic equation \( y = a(x-h)^2 + k \), the directrix is calculated using the formula \( y = k - 1/4a \).

• In our case, \( h = 0 \), \( k = 0 \), and \( a = 1 \), so the directrix equation becomes \( y = -1/4 \).

This implies the directrix is a horizontal line located a quarter unit below the vertex.

The directrix acts as a guidance line to ensure every point on the parabola remains equidistant from both the focus and the directrix, securing the symmetrical curve nature of the parabola.