Problem 23
Question
Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the \(y\) -axis that passes through the point (2,-6)
Step-by-Step Solution
Verified Answer
Answer: The equation of the parabola is \(x = \frac{1}{18}y^2\).
1Step 1: Write the general equation of the parabola
The general equation of a parabola with vertex at the origin and symmetric about the y-axis is given by \(x = ay^2\). Our goal is to find the value of "a".
2Step 2: Plug the given point into the equation
We are given the point (2, -6) which lies on the parabola. Plug this point into the equation \(x = ay^2\), so that x = 2 and y = -6:
\(2 = a(-6)^2\)
3Step 3: Solve for the constant "a"
Now, we simply need to solve the equation from Step 2 for "a":
\(2 = a(-6)^2\)
\(2 = 36a\)
Divide by 36 to isolate "a":
\(a = \frac{1}{18}\)
4Step 4: Write the equation of the parabola
Now that we have found the value of "a", we can write the equation of the parabola with vertex at the origin and symmetric about the y-axis:
\(x = \frac{1}{18}y^2\)
That's our final answer: the equation of the parabola is \(x = \frac{1}{18}y^2\).
Key Concepts
Parabola SymmetrySolving for ConstantsVertex of a ParabolaCoordinate Geometry
Parabola Symmetry
Understanding the symmetry of a parabola is fundamental when studying its properties and equation. A parabola is a mirror-symmetrical, plane curve that is U-shaped. This symmetry indicates that one side of the parabola is a mirror image of the other.
In the context of the given exercise, the parabola is symmetric about the y-axis, which means that for any point on the parabola, its reflection across the y-axis will also be on the parabola. The axis of symmetry can be defined as the line that divides the parabola into two congruent halves. For a parabola symmetric about the y-axis, this line is vertical and can be described by the equation x=0, which corresponds to the y-axis itself.
When the vertex is at the origin (0,0), and the parabola is symmetric about the y-axis, the general equation takes a simpler form: \(x = ay^2\). This equation highlights the direct relationship between x and the square of y, showcasing the parabolic shape's unique characteristics.
In the context of the given exercise, the parabola is symmetric about the y-axis, which means that for any point on the parabola, its reflection across the y-axis will also be on the parabola. The axis of symmetry can be defined as the line that divides the parabola into two congruent halves. For a parabola symmetric about the y-axis, this line is vertical and can be described by the equation x=0, which corresponds to the y-axis itself.
When the vertex is at the origin (0,0), and the parabola is symmetric about the y-axis, the general equation takes a simpler form: \(x = ay^2\). This equation highlights the direct relationship between x and the square of y, showcasing the parabolic shape's unique characteristics.
Solving for Constants
To determine the specific equation of a parabola, we must solve for the constants in the equation. These constants define the precise shape and positioning of the parabola on the coordinate plane.
In our exercise, we have the general form \(x = ay^2\), with 'a' being the constant we need to find. By inserting a known point that lies on the parabola into the equation, we can solve for 'a'. This step is crucial since 'a' affects the parabola's width and direction.
The process is quite straightforward: plug in the x- and y-values from the given point and solve the resulting equation for 'a'. As seen in the solution steps, the point (2, -6) gives us the equation \(2 = a(-6)^2\), which simplifies to \(2 = 36a\). Solving for 'a' yields \(a = \frac{1}{18}\). This value is now the constant that defines our specific parabola.
In our exercise, we have the general form \(x = ay^2\), with 'a' being the constant we need to find. By inserting a known point that lies on the parabola into the equation, we can solve for 'a'. This step is crucial since 'a' affects the parabola's width and direction.
The process is quite straightforward: plug in the x- and y-values from the given point and solve the resulting equation for 'a'. As seen in the solution steps, the point (2, -6) gives us the equation \(2 = a(-6)^2\), which simplifies to \(2 = 36a\). Solving for 'a' yields \(a = \frac{1}{18}\). This value is now the constant that defines our specific parabola.
Vertex of a Parabola
The vertex of a parabola can be thought of as the 'tip' or the highest or lowest point, depending on which way the parabola opens. It's a critical point since it defines the parabola's maximum or minimum value and its location on the coordinate plane.
For the exercise, we are given that the vertex is at the origin (0,0). This significantly simplifies our equation, as it eliminates any horizontal or vertical shifts that would otherwise be present if the vertex was located elsewhere. With the vertex at the origin, our parabola is positioned such that it opens along one of the axis, and its equation follows the standard form associated with this setup, which is a crucial piece of information for solving parabola-related problems in coordinate geometry.
For the exercise, we are given that the vertex is at the origin (0,0). This significantly simplifies our equation, as it eliminates any horizontal or vertical shifts that would otherwise be present if the vertex was located elsewhere. With the vertex at the origin, our parabola is positioned such that it opens along one of the axis, and its equation follows the standard form associated with this setup, which is a crucial piece of information for solving parabola-related problems in coordinate geometry.
Coordinate Geometry
Coordinate geometry, often known as analytic geometry, meticulously combines algebra and geometry to enable the solving of geometric problems. It involves graphing and dealing with curves, such as parabolas, and lines on a coordinate plane.
In coordinate geometry, each point is described by an ordered pair of numbers known as coordinates, representing its position on a plane. For example, the given point in the exercise is (2, -6), where 2 is the x-coordinate, and -6 is the y-coordinate. These coordinates allow us to visualize geometric figures and to formulate their equations.
When dealing with parabolas in coordinate geometry, we focus on features such as the vertex, focus, directrix, and axis of symmetry. These characteristics aid in deriving the general equations that represent the parabolic curves, as we've seen in the exercise example. It is the interplay between algebraic manipulation and geometric interpretation that makes coordinate geometry a powerful tool in mathematics.
In coordinate geometry, each point is described by an ordered pair of numbers known as coordinates, representing its position on a plane. For example, the given point in the exercise is (2, -6), where 2 is the x-coordinate, and -6 is the y-coordinate. These coordinates allow us to visualize geometric figures and to formulate their equations.
When dealing with parabolas in coordinate geometry, we focus on features such as the vertex, focus, directrix, and axis of symmetry. These characteristics aid in deriving the general equations that represent the parabolic curves, as we've seen in the exercise example. It is the interplay between algebraic manipulation and geometric interpretation that makes coordinate geometry a powerful tool in mathematics.
Other exercises in this chapter
Problem 22
Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the right lobe of \(r=\sqrt{\cos 2 \theta}\)
View solution Problem 22
Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate t
View solution Problem 23
Express the following Cartesian coordinates in polar coordinates in at least two different ways. \((1, \sqrt{3})\)
View solution Problem 23
Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the circle \(r=8 \sin \theta\)
View solution