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Chapter 7: Problem 67

Give the focus, directrix, and axis of each parabola. $$x^{2}=16 y$$

Short Answer

Expert verified
Focus: (0, 4), Directrix: y = -4, Axis: x = 0.

Step by step solution

01

Identify the Parabola Form

The given equation is \(x^2 = 16y\), which matches the form \(x^2 = 4py\), where the parabola opens upwards or downwards. Here, \(x\) is squared, indicating a vertical axis of symmetry.
02

Determine the Focus

To find the focus, we need to find \(p\). Given \(x^2 = 4py\), we compare this with our equation \(x^2 = 16y\) and identify that \(4p = 16\). Solving for \(p\), we get \(p = 4\). Thus, the focus is located at \((0, p) = (0, 4)\).
03

Determine the Directrix

The directrix of a parabola is given by \(y = -p\). So with \(p = 4\), the equation of the directrix is \(y = -4\).
04

Identify the Axis of Symmetry

The axis of symmetry for a parabola in the form \(x^2 = 4py\) is the vertical line whose equation is \(x=0\). It is a vertical line along the \(y\)-axis.

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that can open either upward or downward if it lies on a vertical axis or side-to-side along a horizontal axis. Parabolas are a type of conic section, which means they are formed by the intersection of a plane with a right circular cone. In the equation \(x^2 = 16y\), the parabola is said to be vertically oriented, as \(x\) is the squared variable, indicating that the parabola opens upwards or downwards.

Key properties of a parabola include:
  • The vertex, which is the highest or lowest point on the parabola depending on its orientation.
  • The axis of symmetry, which is a line that divides the parabola into two mirror-image halves.
  • The focus and directrix, which help define the parabola's shape and position.
Understanding these properties is crucial to graphing and analyzing the parabolic equations effectively.
Focus and Directrix
The focus and directrix are crucial in defining the exact shape and position of a parabola. For the parabola \(x^2 = 16y\), determining these components relies on understanding the equation鈥檚 standard form. Here we have \(x^2 = 4py\), where \(p\) is a constant that helps locate both the focus and the directrix.

In this example, by comparing \(x^2 = 16y\) with \(x^2 = 4py\), we find that \(4p = 16\), hence \(p = 4\).
  • The focus is a point on the interior of the parabola, and it is located at \((0, p)\), which means for our equation, the focus is at \((0, 4)\).
  • The directrix is a horizontal line that lies outside the parabola, corresponding to \(y = -p\). In this case, the directrix is \(y = -4\).
Every point on the parabola is equidistant from the focus and the directrix, which is a fundamental property that defines its shape.
Axis of Symmetry
The axis of symmetry of a parabola is a vertical or horizontal line that divides the parabolic curve into two equal halves, mirroring each other. This line passes through the vertex of the parabola and is a crucial part of the geometric structure.

For parabolas oriented vertically like \(x^2 = 16y\), the axis of symmetry is vertical. Specifically, it is the line \(x = 0\). This axis runs along the y-axis of the coordinate plane.
  • The axis of symmetry directly affects how the parabola is graphed. Knowing this line helps in perfectly sketching both sides of the parabola.
  • It also determines the direction in which the parabola opens.
Understanding the axis of symmetry helps in predicting the behavior of the parabola and solving problems related to its structure and graph.

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Most popular questions from this chapter

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y^{2}+2 y+1=-2 x+4$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$$

Find an equation of a parabola that satisfies the given conditions. Focus \((-1,2) ;\) vertex \((3,2)\)

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$4(x+9)^{2}-25(y+6)^{2}=100$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y-3)^{2}$$
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