Problem 18 - Algebra 2

Question

Identify the focus and the directrix of the graph of each equation. $$ y=-\frac{1}{8} x^{2} $$

Step-by-Step Solution

Answer

The focus of the given parabolic equation is at (0, -2) and the equation of the directrix is y = 2.

Step 1: Determine the Vertex

In a parabola of the form \(y = ax^2\), the vertex is always at the origin (0,0).

Step 2: Find the Distance from Vertex to Focus and Directrix

The distance, \(d\), from the vertex to the focus (and also from the vertex to the directrix) is calculated as \(d = 1/4|a|\). Here, \(a = -1/8\), so \(d = 1/(4|-1/8|) = 2\).

Step 3: Calculate the Coordinates of the Focus

The focus of a parabola \(y = ax^2\) is at (h, k-d) if \(a < 0\), and at (h, k+d) if \(a > 0\). Since a is negative in this case, we subtract the distance from the y-coordinate of the vertex to get the focus. So, Focus = (0, 0-2) = (0, -2).

Step 4: Write the Equation of the Directrix

The directrix of a parabola \(y = ax^2\) is \(y = k + d\) if \(a < 0\), and \(y = k - d\) if \(a > 0\). Since \(a\) is negative in this case, we add the distance to the y-coordinate of the vertex to get the directrix. So, Directrix: \(y = 0 + 2 = 2\).

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that is formed by a quadratic function of the form \( y = ax^2 + bx + c \), where \( a eq 0 \). This distinctive shape is widely seen in various fields such as physics, engineering, and economics. In a parabola, every point is equidistant from a fixed point called the "focus" and a line known as the "directrix." This property is what contributes to its unique shape and symmetry.

Key characteristics of a parabola include:

• Vertex: The highest or lowest point of the parabola depending on whether it opens upwards or downwards.
• Axis of symmetry: A vertical line through the vertex that divides the parabola into two mirror-image halves.
• Focus: A point inside the parabola which determines its "direction," situated \(d\) distance away from the vertex.
• Directrix: A line outside the parabola also \(d\) distance away from the vertex, perpendicular to the axis of symmetry.

Understanding these elements can help in graphing and analyzing parabolas effectively in different contexts.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally represented as \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). These equations are crucial in algebra and can be solved using various methods such as factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Quadratic equations give rise to parabolas when plotted on a graph. The coefficient \(a\):

• determines the opening direction of the parabola. If \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.
• affects the "width" of the parabola. Larger values of \(|a|\) make the parabola narrower, while smaller values make it wider.

Analyzing quadratic equations involves identifying the vertex, axis of symmetry, and direction of opening, which allows for understanding how changes in \(a\), \(b\), and \(c\) impact the graph. Exploring real-world phenomena often relies on solving and interpreting these quadratic equations.

Vertex

The vertex is a critical feature of a parabola that marks its turning point. In the standard form of quadratic equations \( y = ax^2 + bx + c \), the position of the vertex can be ascertained through various approaches. The vertex form \( y = a(x-h)^2 + k \) directly reveals it as \((h, k)\).

For standard form equations, the x-coordinate of the vertex can be calculated using \( x = -\frac{b}{2a} \), and the y-coordinate is found by substituting this x-value back into the equation. This yields the vertex \((x, y)\), which serves as a guide for graphing the parabola. The parabola is symmetrical around the vertical line through this vertex, known as the axis of symmetry.

In particular cases, like the equation \( y = -\frac{1}{8}x^2 \), the vertex is at the origin \((0, 0)\). This results in equal distances to the focus and directrix, maintaining a neat symmetry in the graph and helping predict the parabola's behavior. Recognizing the vertex is vital for understanding and manipulating the shape of the parabola efficiently.