Problem 59 - Algebra 2

Question

Modeling Draw a cross section of a parabolic mirror modeled by the equation \(y=0.002323 x^{2}\) .

Step-by-Step Solution

Answer

The cross-section of a parabolic mirror modeled by the equation \(y=0.002323x^{2}\) is a parabola that opens upwards with its vertex at the origin.

Step 1: Understanding the equation

The equation given is \(y=0.002323x^{2}\). This is a quadratic equation in the form of \(y=ax^{2}\), where a is the coefficient of \(x^{2}\). Here, \(a=0.002323\), which is a positive number. In the graphical representation of this type of equations, when a is positive, the parabola opens upwards.

Step 2: Identifying the vertex of the parabola

The vertex of the parabola given by the equation \(y=ax^{2}\) is at the origin of the coordinate system \((0,0)\). This is because there is no constant or linear term in the equation, which would shift the parabola up/down or left/right respectively.

Step 3: Plotting the parabola

Now, plot the parabola using the origin \((0,0)\) as the vertex. Construct the parabola by plotting points. For instance, substituting \(x=100\) into the equation to find the corresponding y-coordinate, \(y=0.002323(100)^{2} = 23.23\). So, the point \((100,23.23)\) lies on the parabola. Similarly, find some more points lying on the parabola by choosing different x-values and locating the corresponding y-values by means of the equation. Join these points to get the parabolic shape. The cross section of the parabolic mirror will be symmetric about the y-axis as the coefficient of \(x^{2}\) is positive.

Step 4: Interpreting the graph

The created graph represents the cross-section of a parabolic mirror. Each point on the parabola corresponds to a point on the actual mirror. This model can be used to analyze the properties of the mirror, such as its focus and directrix.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is one of the foundational elements of algebra. It can be expressed in the standard form:

• \( ax^2 + bx + c = 0 \)

In the equation given, \( y = 0.002323x^2 \), the form is slightly different because it lacks the \( bx \) and \( c \) terms, making it a simplified quadratic equation. This form, \( y = ax^2 \), is a pure quadratic, where \( a \) determines the parabola's opening direction and width. Since \( a = 0.002323 \) is positive, the parabola opens upwards.
Quadratic equations define parabolic shapes, which have unique properties such as symmetry, a vertex, and a consistent curve. Understanding these allows you to graph them effectively and apply them to real-world situations like modeling mirrors or satellite dishes.

Exploring the Vertex of a Parabola

The vertex is the central point of a parabola and is often considered its peak or trough. For the equation \( y = 0.002323x^2 \), the vertex is located at the origin \( (0,0) \).
This occurs because the equation lacks any additional terms that would shift the parabola along the x or y axes.

• If there were a \( bx \) term, the vertex would move left or right.
• A constant \( c \) would move it up or down.

Knowing the location of the vertex is crucial for graphing the parabola, as it serves as a key reference point from which the rest of the curve develops symmetrically on either side. This symmetry makes it easier to predict and plot other points on the graph.

Understanding the Coordinate System

In mathematics, the coordinate system is a grid that helps visualize equations by positioning them in a space defined by two axes: x (horizontal) and y (vertical). The origin \((0,0)\) is the intersection of these axes.
For a parabola defined by \( y = 0.002323x^2 \), plotting begins at the origin. The coordinates describe where points fall on the graph, enabling precise mapping of the parabolic shape.

• Choose x-values, plug them into the equation, and solve for y to get coordinates like \((100, 23.23)\).
• Plot these points using the grid, connecting them to form the smooth curve of the parabola.

The coordinate system allows easy identification of symmetry and distance, helping analyze parabola dimensions efficiently.

Cross-Section Modeling with Parabolas

Cross-section modeling with parabolas allows us to represent real-world objects such as mirrors.
The equation \( y = 0.002323x^2 \) models a parabolic mirror's cross-section, illustrating how the curve translates into a physical structure. This implies that:

• Every point on the graph corresponds to a point on the mirror.
• The symmetry ensures uniformity across the mirror's surface.

Such models are not only useful in optics but in fields like architecture and automotive design. Analyzing the parabola's focus and directrix helps in designing dishes and other reflective surfaces to optimize performance.