Problem 26 - Algebra and Trigonometry

Question

$$ \text { Use a graphing device to graph the parabola. } $$ $$ 8 y^{2}=x $$

Step-by-Step Solution

Answer

The parabola opens to the right, vertex at (0,0), focus at (2,0), and directrix x=-2.

Step 1: Understand the Equation

The given equation is \(8y^2 = x\). This is a quadratic equation in terms of \(y\), indicating that it represents a parabola. In this form, it suggests the parabola opens sideways.

Step 2: Convert to Standard Form

Rearrange the equation to match the standard form of a sideways parabola \((y - k)^2 = 4p(x - h)\). Divide both sides by 8 to get \(y^2 = \frac{1}{8}x\). This form shows the parabola is centered at \((0,0)\) with \(8 = 4p\) hence \(p = 2\).

Step 3: Identify Key Features

The vertex of the parabola is \((0,0)\), the focus is \((2,0)\), and the directrix is the line \(x = -2\). This shows that the parabola opens to the right.

Step 4: Graph the Parabola

Plot the vertex, focus, and the directrix on a graph. Draw the parabola opening towards the positive x-direction, making sure it is symmetric about the x-axis.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a right circular cone with a plane. The journey through conic sections unfolds a mathematical masterpiece comprising circles, ellipses, parabolas, and hyperbolas.
Among these, parabolas emerge as fascinating shapes defined by a single quadratic term. These curves can open upwards, downwards, or even sideways depending on their equation form. Conic sections have numerous applications in physics, engineering, and astronomy, where they describe orbits or the path of projectiles.
Understanding conic sections aids in visualizing and creating graphs of quadratic functions, enhancing problem-solving skills in multiple scenarios.

Graphing Parabolas

Graphing a parabola involves finding its vertex, axis of symmetry, focus, directrix, and direction of opening. The process helps visualize how a parabola behaves on a coordinate plane.
To graph a parabola effectively:

• Identify whether it opens vertically or horizontally based on the equation.
• Locate the vertex, which is the turning point of the curve.
• Determine the focus and directrix, guiding the curve's shape and position.
• Observe the axis of symmetry, a line through the vertex dividing the parabola into two identical halves.

Drawing these on a graph gives a complete picture, allowing for analysis of the trajectory or system the parabola models.

Focus and Directrix

A parabola is uniquely defined by its focus and directrix. These elements help understand its geometric properties and nature.
The focus of a parabola is a fixed point from which distances are measured to form the curve. Meanwhile, the directrix is a fixed line that maintains these distances. The parabola comprises points equidistant from both the focus and the directrix.
This property is crucial in applications like satellite dishes, reflecting signals toward a receiver placed at the focus. When graphing, mark the focus and draw the directrix to ensure the parabola accurately represents the intended scenario.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They take the standard form as either \( ax^2 + bx + c = 0 \) for vertical parabolas, or \( ay^2 + by + c = x \) for horizontal parabolas.
These equations model various problems, from projectile motion to maximizing areas. Solutions to quadratic equations, called the roots or x-intercepts, reveal key points on the graph where the parabola crosses the x-axis.
Understanding how to manipulate and solve these equations is essential in algebra. It acts as a gateway to exploring more complex mathematical concepts and practical applications.

Standard Form of Parabola

The standard form of a parabola provides a concise way to identify its properties immediately. For a vertical parabola, the standard form is \( (x - h)^2 = 4p(y - k) \), and for a horizontal parabola, it is \( (y - k)^2 = 4p(x - h) \).
In both, \((h,k)\) represents the vertex, with \(p\)indicating the distance from the vertex to the focus or directrix. Converting any quadratic equation to its standard form simplifies finding these features.
The standard form is not only a mathematical tool but also an educational one, aiding students in visualizing and predicting the behavior of parabolas, ensuring a better grasp of their graphical representations.