Problem 32 - Contemporary Precalculus

Question

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y=2 x^{2}$$

Step-by-Step Solution

Answer

Answer: The latus rectum of the parabola \(y=2x^{2}\) is the line segment with endpoints \((-4, \frac{1}{8})\) and \((4, \frac{1}{8})\).

Step 1: Determine the standard form of the parabola

We have the equation \(y=2x^{2}\). This is already in the standard form for a parabola with a vertical axis of symmetry, which is given by: $$y=a x^{2}$$ where \(a\) is the coefficient of the squared term (\(x^{2}\)). In this case, \(a=2\).

Step 2: Find the vertex of the parabola

For a parabola in the form \(y=ax^{2}\), the vertex is located at the origin \((0,0)\). This is because the parabola is symmetric with respect to the y-axis. In this case, the vertex is \((0,0)\).

Step 3: Find the focus of the parabola

We can find the focus of the parabola using the formula $$\frac{1}{4a}$$ In this case, \(a=2\), so the distance from the vertex to the focus is: $$\frac{1}{4(2)} = \frac{1}{8}$$ Since the parabola opens upwards, the focus is located \(\frac{1}{8}\) units above the vertex. So, the focus is at the point \((0,\frac{1}{8})\).

Step 4: Find the latus rectum

The latus rectum is the line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. The length of the latus rectum is given by the formula \(|4a|\), where \(a\) is the coefficient of the squared term. In this case, \(a=2\). So, the length of the latus rectum is: $$|4(2)| = 8$$ Since the latus rectum is centered at the focus and is \(8\) units long, the endpoints of the latus rectum are given by: \((x,\frac{1}{8})\) such that: $$x=-4,0,4$$ So, the latus rectum of the parabola \(y=2x^{2}\) is the line segment with endpoints \((-4, \frac{1}{8})\) and \((4, \frac{1}{8})\).

Key Concepts

Standard Form of a Parabola

When studying parabolas, understanding their standard form is essential. The standard form of a parabola with a vertical axis of symmetry is represented by the equation
\[ y = ax^{2} + bx + c \]
where the variables 'a', 'b', and 'c' are coefficients that shape the parabola. If the 'a' value is positive, the parabola opens upwards; if negative, it opens downwards. This equation showcases how the parabola relates to the Cartesian coordinate system, where '(x, y)' represents any point on the curve.

For instance, the equation given in the exercise, \[ y = 2x^{2} \], is in the standard form, indicating a parabola that opens upwards (since 'a' is positive) with a coefficient 'a' equal to 2. This form is useful for determining several characteristics of the parabola, including its width, orientation, and the location of its vertex, focus, and latus rectum.

Vertex of a Parabola

The vertex of a parabola is a critical concept in its geometry as it represents the highest or lowest point on the curve, depending on its orientation. In the standard form \[ y = ax^{2} + bx + c \], the vertex can be found using the formula
\[(-\frac{b}{2a}, ax^{2} + bx + c)\]
for parabolas that open up or down. In cases where the parabola is in the form \[ y = ax^{2} \] or \[ x = ay^{2} \], the vertex will be at the origin (0, 0), because it is symmetric about either the x-axis or y-axis.

For the given exercise, with the parabola equation \[ y = 2x^{2} \], the vertex is found at the origin. This is important when calculating other features, such as the focus or latus rectum.

Focus of a Parabola

The focus of a parabola is a fixed point from which each point on the parabola is equidistant from a corresponding point on the directrix, thereby defining its shape. For a parabola in the form of \[ y = ax^{2} \], the focus lies on the axis of symmetry and can be found using the formula
\[ (0, \frac{1}{4a}) \]
where 'a' is the coefficient from the equation. The focus is not merely an abstract concept; it has practical applications in optics and engineering, as parabolic shapes are used to reflect light or sound waves to this focal point.

In our exercise where we have \[ y = 2x^{2} \], the focus is \[ (0, \frac{1}{8}) \]. It is located \[ \frac{1}{8} \] units above the vertex since our parabola opens upwards. This information is valuable when determining the locus of points that form the parabola and for evaluating properties like the latus rectum.