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Chapter 13: Problem 29

Identify whether each equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of each equation. \(x^{2}+y^{2}=16\)

Short Answer

Expert verified
The equation represents a circle with radius 4.

Step by step solution

01

Review the General Forms

To determine the type of conic section an equation represents, we start by recalling some key forms: - A **circle** is given by the equation \(x^2 + y^2 = r^2\), where \(r\) is the radius.- An **ellipse** is of the form \(ax^2 + by^2 = 1\), where \(a\), \(b\) have the same sign but are different.- A **parabola** is usually expressed as \(y = ax^2\) or \(x = ay^2\), having only one squared term.- A **hyperbola** has the form \(ax^2 - by^2 = 1\), where \(a\) and \(b\) have opposite signs.
02

Identify the Equation Type

Given equation is \(x^2 + y^2 = 16\). This matches with the general form of a circle, \(x^2 + y^2 = r^2\), where \(r^2 = 16\). Therefore, this is the equation of a circle with radius 4.
03

Sketch the Graph

To sketch the equation \(x^2 + y^2 = 16\), we recognize it as a circle centered at the origin (0,0) with radius 4. Plot the center of the circle at the origin, and draw a circle that passes through the points (4,0), (-4,0), (0,4), and (0,-4) on the Cartesian plane.

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Key Concepts

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

Circle Equation
A circle equation is a special type of algebraic expression that defines a circle on a coordinate plane. It is one of the basic forms of conic sections. The standard form of a circle's equation is given by:- \(x^2 + y^2 = r^2\)
  • Center: The center of the circle is at the origin, which is the point (0,0) in this case.
  • Radius: The radius is determined from the equation. In our exercise, since we have \(x^2 + y^2 = 16\), the radius is \(r = \sqrt{16} = 4\).
This equation means that any point \((x, y)\) on the perimeter of the circle is exactly 4 units away from the center of the circle. Notice that the equation includes both \(x\) and \(y\) as squared terms without coefficients other than one, which is a key identifier of a circle.
Graphing Equations
Graphing equations is a fundamental skill in mathematics. For conic sections, graphing is essential for visualization and understanding the shapes described by algebraic equations. When graphing the circle defined by \(x^2 + y^2 = 16\), one can follow a straightforward method:- Recognize the equation form: Identify that it represents a circle.- Determine the circle's center and radius as discussed.Now, graph the circle on a Cartesian plane:
  • Locate the center of the circle at the origin (0,0).
  • From the center, move 4 units in all cardinal directions (right, left, up, down) to identify points (4,0), (-4,0), (0,4), and (0,-4).
  • Connect these points smoothly to form a perfect circle.
By visualizing this graph, you can see that all points on the circle maintain a consistent distance, known as the radius, from the center point.
Identifying Conic Sections
Conic sections are curves obtained by intersecting a plane with a double-napped cone. Each distinct form or pattern is identified by specific algebraic equations:
  • Circle: Equal coefficients and no mixed terms, i.e., \(x^2 + y^2 = r^2\).
  • Ellipse: Coefficients same sign but different values in the form \(ax^2 + by^2 = 1\).
  • Parabola: A single squared term, like \(y = ax^2\) or \(x = ay^2\).
  • Hyperbola: Coefficients have opposite signs, such as \(ax^2 - by^2 = 1\).
Identifying the type of conic section is a crucial first step in graphing and understanding the geometric properties of the expression. With practice, recognizing the patterns between these forms becomes intuitive, aiding in both problem-solving and visualization of mathematical concepts.

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Most popular questions from this chapter

Rationalize each denominator and simplify, if possible. See Section 10.5 $$ \frac{4 \sqrt{7}}{\sqrt{6}} $$

The graph of each equation is an ellipse. Determine which distance is longer. The distance between the \(x\) -intercepts or the distance between the \(y\) -intercepts. How much longer? \(x^{2}+4 y^{2}=36\)

Perform the indicated operations. \(-5 x^{2}+x^{2}\)

Graph each inequality in two variables. $$ y<2 x-1 $$

For the exercises below, see the Concept Check in this section. Without graphing. how can you tell that the graph of \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\) do not have any points of intersection?
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