Graph The Circle.$4x^2 + 24x + 4y^2 - 12y = -29$

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Introduction

Graphing a circle involves converting the given equation into the standard form of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this article, we will graph the circle given by the equation 4x^2 + 24x + 4y^2 - 12y = -29.

Step 1: Convert the Equation to Standard Form

To convert the given equation into the standard form, we need to complete the square for both x and y terms.

4x^2 + 24x + 4y^2 - 12y = -29

First, let's factor out the constants of the squared terms:

4(x^2 + 6x) + 4(y^2 - 3y) = -29

Next, we add and subtract the square of half the coefficient of x and y inside the parentheses:

4(x^2 + 6x + 9 - 9) + 4(y^2 - 3y + 9/4 - 9/4) = -29

Now, we can rewrite the equation as:

4(x^2 + 6x + 9) - 36 + 4(y^2 - 3y + 9/4) - 9/4 = -29

Simplifying the equation, we get:

4(x + 3)^2 + 4(y - 3/2)^2 = 36 + 9/4 + 29

Combining the constants on the right-hand side, we get:

4(x + 3)^2 + 4(y - 3/2)^2 = 36 + 37/4

To add the fractions on the right-hand side, we need a common denominator, which is 4. So, we can rewrite the equation as:

4(x + 3)^2 + 4(y - 3/2)^2 = (144 + 37)/4

Simplifying the fraction, we get:

4(x + 3)^2 + 4(y - 3/2)^2 = 181/4

Now, we can rewrite the equation in the standard form:

(x + 3)^2 + (y - 3/2)^2 = 181/16

Step 2: Identify the Center and Radius

Comparing the equation with the standard form, we can see that the center of the circle is (-3, 3/2) and the radius is √(181/16).

Step 3: Graph the Circle

To graph the circle, we can use the center and radius to draw a circle on the coordinate plane.

Conclusion

In this article, we graphed the circle given by the equation 4x^2 + 24x + 4y^2 - 12y = -29. We converted the equation into the standard form, identified the center and radius, and graphed the circle on the coordinate plane.

Key Takeaways

  • To graph a circle, we need to convert the given equation into the standard form.
  • The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
  • We can use the center and radius to draw a circle on the coordinate plane.

Frequently Asked Questions

  • Q: How do I convert the equation of a circle into the standard form? A: To convert the equation of a circle into the standard form, we need to complete the square for both x and y terms.
  • Q: What is the standard form of a circle? A: The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
  • Q: How do I graph a circle on the coordinate plane? A: To graph a circle on the coordinate plane, we can use the center and radius to draw a circle.
    Graphing the Circle: A Q&A Guide =====================================

Introduction

In our previous article, we graphed the circle given by the equation 4x^2 + 24x + 4y^2 - 12y = -29. We converted the equation into the standard form, identified the center and radius, and graphed the circle on the coordinate plane. In this article, we will answer some frequently asked questions about graphing circles.

Q&A

Q: What is the standard form of a circle?

A: The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Q: How do I convert the equation of a circle into the standard form?

A: To convert the equation of a circle into the standard form, you need to complete the square for both x and y terms. This involves adding and subtracting the square of half the coefficient of x and y inside the parentheses.

Q: What is the center of a circle?

A: The center of a circle is the point (h, k) in the standard form of a circle, where (x - h)^2 + (y - k)^2 = r^2.

Q: How do I find the center of a circle?

A: To find the center of a circle, you need to look at the equation in the standard form and identify the values of h and k.

Q: What is the radius of a circle?

A: The radius of a circle is the value of r in the standard form of a circle, where (x - h)^2 + (y - k)^2 = r^2.

Q: How do I find the radius of a circle?

A: To find the radius of a circle, you need to look at the equation in the standard form and identify the value of r.

Q: How do I graph a circle on the coordinate plane?

A: To graph a circle on the coordinate plane, you can use the center and radius to draw a circle. You can start by plotting the center of the circle and then drawing a circle with the given radius.

Q: What are some common mistakes to avoid when graphing a circle?

A: Some common mistakes to avoid when graphing a circle include:

  • Not converting the equation into the standard form
  • Not identifying the center and radius correctly
  • Not using the correct values for h and k
  • Not drawing the circle with the correct radius

Q: How do I check my work when graphing a circle?

A: To check your work when graphing a circle, you can:

  • Verify that the equation is in the standard form
  • Check that the center and radius are correct
  • Draw the circle and check that it matches the equation
  • Use a calculator or graphing software to check the graph

Conclusion

Graphing a circle involves converting the equation into the standard form, identifying the center and radius, and graphing the circle on the coordinate plane. By following these steps and avoiding common mistakes, you can accurately graph a circle. If you have any further questions or need additional help, please don't hesitate to ask.

Key Takeaways

  • The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
  • To convert the equation of a circle into the standard form, you need to complete the square for both x and y terms.
  • The center of a circle is the point (h, k) in the standard form of a circle.
  • The radius of a circle is the value of r in the standard form of a circle.
  • To graph a circle on the coordinate plane, you can use the center and radius to draw a circle.

Frequently Asked Questions

  • Q: How do I convert the equation of a circle into the standard form? A: To convert the equation of a circle into the standard form, you need to complete the square for both x and y terms.
  • Q: What is the standard form of a circle? A: The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
  • Q: How do I find the center of a circle? A: To find the center of a circle, you need to look at the equation in the standard form and identify the values of h and k.
  • Q: What is the radius of a circle? A: The radius of a circle is the value of r in the standard form of a circle.
  • Q: How do I graph a circle on the coordinate plane? A: To graph a circle on the coordinate plane, you can use the center and radius to draw a circle.