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Chapter 11: Problem 22

Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation. Focus at (0,2)\(;\) vertex at (0,0)

Short Answer

Expert verified
The equation is \( x^2 = 8y \). The points defining the latus rectum are (4,2) and (-4,2).

Step by step solution

01

Identify the vertex and focus coordinates

The vertex is at (0,0) and the focus is at (0,2). The parabolic equation can be determined using these points.
02

Determine the value of p

The distance from the vertex to the focus is equal to p. In this case, the focus is 2 units above the vertex, so \( p = 2 \).
03

Use the vertex form of a parabolic equation

For a parabola with vertex at (0,0) and opening upwards, the equation is: \[ x^2 = 4py \] Substitute \( p \) with 2: \[ x^2 = 8y \].
04

Find the points that define the latus rectum

The latus rectum is a line segment perpendicular to the axis of symmetry and passing through the focus. In this case, it is vertical since the axis of symmetry is the y-axis. The length of the latus rectum is \( 4p \), which is 8. The points are found by moving 4 units left and right of the focus (0,2): The points are (4,2) and (-4,2).
05

Graph the equation

To graph the equation \( x^2 = 8y \), plot the vertex at (0,0), the focus at (0,2), and the two points defining the latus rectum at (4,2) and (-4,2). Then, draw the parabola opening upwards passing through all these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex
The vertex of a parabola is the point where the parabola changes direction. It serves as the minimum or maximum point depending on the parabola's orientation. In this exercise, the vertex is at (0,0). This is crucial because it acts as the central point from which other key features of the parabola are measured. For parabolas in vertex form, the equation looks like: \( y = ax^2 + bx + c \)
Since our vertex is at (0,0) and the equation is centered around this point, our simplified equation becomes \( x^2 = 4py \). Knowing the vertex helps you place the parabola correctly on a graph.
focus
The focus of a parabola is a point from which distances are measured to define the curve. For a parabola opening upward, this point lies along the y-axis from the vertex. The distance from the vertex to the focus is denoted by 'p'.
In this example, the focus is at (0,2). This means that the distance 'p' is 2 units. The standard form of a parabola's equation utilizes this distance to shape the curve correctly. Here, the equation becomes: \[ x^2 = 8y \].
Understanding the focus is vital because it ensures the parabola's curvature aligns properly with the vertex, creating an accurate graph.
latus rectum
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry, passing through the focus. It spans the width of the parabola, aiding in understanding its 'width'. Its length is given by the formula \( 4p \).
In our case, 'p' is 2, so the latus rectum length is 8. The points defining this line segment are located symmetrically around the y-axis. From the focus (0,2), move 4 units left to (-4,2) and 4 units right to (4,2). These points are crucial in plotting the curvature of the parabola accurately and visualizing its width.
graphing parabolas
Graphing parabolas involves plotting all the key points and understanding their relationships. Follow these easy steps:
  • Identify and plot the vertex: (0,0) in our example.
  • Locate and mark the focus: (0,2).
  • Determine the points from the latus rectum: (4, 2) and (-4, 2).
  • Sketch the parabola opening upwards, passing through these points.

The vertex represents the lowest point, the focus determines the curve's direction, and the latus rectum dictates its width. By following these steps, you can visualize the parabola accurately. Consistent practice helps in mastering graphing parabolas.

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