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Chapter 13: Problem 48

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$2 x^{2}+2 y^{2}=\frac{1}{2}$$

Short Answer

Expert verified
It's a circle with center (0,0) and radius \(\frac{1}{2}\).

Step by step solution

01

Simplify the Equation

Start by simplifying the given equation. The equation is \(2x^2 + 2y^2 = \frac{1}{2}\). Divide everything by 2 to get \(x^2 + y^2 = \frac{1}{4}\).
02

Recognize the Equation Type

The equation \(x^2 + y^2 = \frac{1}{4}\) is in the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\). This indicates the graph is a circle.
03

Determine the Center of the Circle

In the simplified equation \((x - 0)^2 + (y - 0)^2 = \frac{1}{4}\), the center of the circle \((h,k)\) is \((0,0)\).
04

Calculate the Radius of the Circle

The equation \((x - 0)^2 + (y - 0)^2 = \frac{1}{4}\) shows that \(r^2 = \frac{1}{4}\). Therefore, the radius \(r\) is \(\sqrt{\frac{1}{4}} = \frac{1}{2}\).
05

Sketch the Graph

Sketch a circle with center at the origin \((0,0)\) and radius \(\frac{1}{2}\). The circle will be small and centered around the origin.

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Key Concepts

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

Center of the circle
In a circle's equation, identifying the center is straightforward, given that the equation is in standard form. For our equation, we simplified it to \[(x - 0)^2 + (y - 0)^2 = \frac{1}{4}.\]This clearly points out that the center of the circle, \((h, k)\), is at \((0, 0)\).
  • The terms \(x - h\) and \(y - k\) in the standard form \((x - h)^2 + (y - k)^2 = r^2\) help us determine shifts along the x- and y-axis.
  • Here,聽\(h = 0\) and \(k = 0\); thus, the circle is centered at the origin.
Identifying the center is vital as it assists with positioning the circle on the coordinate plane. This knowledge is a stepping stone to accurately sketching the graph.
Radius of a circle
The radius of a circle is the distance from the center to any point on the circle. Our simplified equation, \[(x - 0)^2 + (y - 0)^2 = \frac{1}{4},\] provides explicit information about the radius.
  • We see that \(r^2 = \frac{1}{4}\), thus \(r = \sqrt{\frac{1}{4}} = \frac{1}{2}.\)
  • This means the circle has a small radius of half a unit.
Understanding the circle's radius is crucial because it defines the physical size of your circle on the graph. It determines how large your circle will appear when plotted.
Standard form of a circle
To graph a circle, we often use its standard form, which is \((x - h)^2 + (y - k)^2 = r^2.\) This expression is incredibly intuitive for identifying both the center and the radius.
  • Comparison: Comparing our equation \(x^2 + y^2 = \frac{1}{4}\) with the standard form confirms it's a circle centered at \((0, 0)\) with a radius \(\frac{1}{2}.\)
  • This form is handy as it clearly shows circle properties, aiding in straightforward calculations for graphing purposes.
Using the standard form is advantageous because it simplifies the graphing process, making it easier for students to understand and visualize the geometric shape.
Graph sketching techniques
When sketching the graph of a circle, follow systematic steps to ensure accuracy and clarity.
  • Identify key elements: Start by verifying the center and the radius based on the standard form. For this circle, the center is \((0, 0)\) and the radius is \(\frac{1}{2}.\)
  • Plot the center: Place a point at \((0, 0)\) for the circle鈥檚 core.
  • Draw the circle: Using the radius \(\frac{1}{2}\), draw a small circle around the point, ensuring all points are \(\frac{1}{2}\) units away from the center.
By following these steps, your graph will accurately represent the circle's mathematical description. Developing your graph sketching skills will improve your ability to visualize equations and understand their geometric representations.

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Most popular questions from this chapter

Write an equation of the circle with the given center and radius. See Example 7. $$ (0,-6) ; \sqrt{2} $$

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$x=-(y-1)^{2}$$

Solve. The sum of the squares of two numbers is \(130 .\) The difference of the squares of the two numbers is \(32 .\) Find the two numbers.

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? \(x^{2}+4 y^{2}=36\)

A bridge constructed over a bayou has a supporting arch in the shape of a parabola. Find an equation of the parabolic arch if the length of the road over the arch is 100 meters and the maximum height of the arch is 40 meters.
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