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Chapter 0: Problem 9

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}+y^{2}=4\)

Short Answer

Expert verified
The graph is a circle centered at the origin with a radius of 2.

Step by step solution

01

Identify the Equation Type

The given equation is \(x^2 + y^2 = 4\). This is an equation of a circle where the general form is \(x^2 + y^2 = r^2\). Therefore, this represents a circle centered at the origin \((0,0)\) with radius \(r = \sqrt{4} = 2\).
02

Check for Symmetries

To check for symmetry, test the equation for invariance under specific transformations. For a circle equation, it is symmetric with respect to the x-axis, y-axis, and the origin by nature. Substituting \(x\) with \(-x\), \(y\) with \(-y\), or both does not change the equation: \((-x)^2 + y^2 = 4\) and \(x^2 + (-y)^2 = 4\). Thus, it is symmetric across all axes.
03

Find the x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\).\[x^2 + 0^2 = 4\]\[x^2 = 4\]\[x = \pm 2\]This gives the intercepts as \((2, 0)\) and \((-2, 0)\).
04

Find the y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation and solve for \(y\).\[0^2 + y^2 = 4\]\[y^2 = 4\]\[y = \pm 2\]This gives the intercepts as \((0, 2)\) and \((0, -2)\).
05

Sketch the Graph

The graph is a circle centered at the origin \((0, 0)\) with radius \(2\). Use the intercepts found at points \((2, 0)\), \((-2, 0)\), \((0, 2)\), and \((0, -2)\) to aid in sketching. The circle encompasses all these points and is symmetric about both axes. The graph should be a full circle touching each of these intercepts.

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Key Concepts

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

Graphical Symmetry
Graphical symmetry is a fascinating concept in mathematics that can significantly simplify the process of graphing equations. A symmetric graph remains unchanged when reflected over certain axes or through certain points. In our example, the equation of a circle, given by \(x^2 + y^2 = 4\), is inherently symmetric.
  • **X-axis symmetry**: Replace \(y\) with \(-y\). Our equation becomes \(x^2 + (-y)^2 = 4\), which is still the same. This means that the graph is symmetric about the x-axis.
  • **Y-axis symmetry**: Replace \(x\) with \(-x\). The equation then is \((-x)^2 + y^2 = 4\). Again, it looks the same, indicating y-axis symmetry.
  • **Origin symmetry**: Replace \(x\) with \(-x\) and \(y\) with \(-y\). This transformation still leaves our equation unchanged, confirming that the circle is symmetric about the origin.
Due to these symmetries, the circle looks identical in all four quadrants. Understanding symmetry can speed up graph drawing and offers deeper insights into the structure of mathematical models.
X-Intercepts
Finding the x-intercepts of a circle involves setting the y-value to zero and solving for x. For the circle with equation \(x^2 + y^2 = 4\), find the x-intercepts by setting \(y = 0\). This simplifies the equation to \(x^2 = 4\). Solving this gives us \(x = 2\) and \(x = -2\).
  • The x-intercepts are found at the points \((2,0)\) and \((-2,0)\).
X-intercepts are the points where the circle crosses the x-axis. They tell us that part of the circle lies in the positive and negative x-regions. Recognizing these intercepts is crucial for accurately sketching the location of the circle in the coordinate plane.
Y-Intercepts
To find the y-intercepts of the circle, we need to set x to zero and solve for y. Using the equation \(x^2 + y^2 = 4\), setting \(x = 0\) simplifies it to \(y^2 = 4\). Solving this gives \(y = 2\) and \(y = -2\).
  • This means the y-intercepts occur at the points \((0, 2)\) and \((0, -2)\).
The y-intercepts are where the circle intersects the y-axis. These points indicate the spread of the circle in the vertical direction and help in drawing the complete circle accurately.
Radius of a Circle
The radius of a circle is a fundamental measure that dictates its size. In our equation of a circle, \(x^2 + y^2 = r^2\), the number in place of \(r^2\) gives us important information. For the given equation \(x^2 + y^2 = 4\), the radius \(r\) is calculated as the square root of 4, which equals 2.
  • The radius forms a straight line from the center of the circle \((0,0)\) to any point on the circle.
  • In our circle, every point is exactly 2 units away from the center in any direction. This forms a perfect round shape.
Understanding the radius allows us to accurately construct the circle on a graph. It also helps to comprehend the spatial relationship and size of the circle relative to other geometrical figures or coordinate points in a given problem.

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Most popular questions from this chapter

Sketch the graph of \(G(t)=t-[t]\)

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Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=(x-1)(x-2)(x-3)\)
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