Problem 27 - Algebra A Combined Function

Question

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}+2 x+12 y-12=0 $$

Step-by-Step Solution

Answer

Center: \((-1, -6)\), Radius: 7

Step 1: Identify the Standard Form of a Circle

The equation of a circle in standard form is \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center and \(r\) is the radius. Our goal is to rewrite the given equation in this form.

Step 2: Rearrange the Given Equation

The given equation is \[ x^2 + y^2 + 2x + 12y - 12 = 0 \].First, move the constant to the other side of the equation:\[ x^2 + y^2 + 2x + 12y = 12 \].

Step 3: Complete the Square for the x-terms

Consider the x-term expression \[ x^2 + 2x \].To complete the square, take half of the coefficient of x (2), square it, and adjust the equation:\[ (x + 1)^2 = x^2 + 2x + 1 \].Add and subtract 1 on the left side:\[ x^2 + 2x = (x + 1)^2 - 1 \].

Step 4: Complete the Square for the y-terms

Consider the y-term expression \[ y^2 + 12y \].To complete the square, take half of the coefficient of y (12), square it and adjust the equation:\[ (y + 6)^2 = y^2 + 12y + 36 \].Add and subtract 36 on the left side:\[ y^2 + 12y = (y + 6)^2 - 36 \].

Step 5: Substitute Completed Squares Back Into Equation

Substitute the completed squares from Steps 3 and 4 into the equation:\[ (x + 1)^2 - 1 + (y + 6)^2 - 36 = 12 \].

Step 6: Simplify the Equation to Standard Form

Combine all constant terms:\[ (x + 1)^2 + (y + 6)^2 = 12 + 1 + 36 = 49 \].Now the equation is in standard form:\[ (x + 1)^2 + (y + 6)^2 = 49 \].

Step 7: Identify the Center and Radius

The equation \((x + 1)^2 + (y + 6)^2 = 49\) is now in the standard form of a circle:- The center \((h, k)\) is \((-1, -6)\).- The radius \(r\) is \(\sqrt{49} = 7\).

Step 8: Graphing the Circle

To graph the circle, plot the center at \((-1, -6)\) on a coordinate plane. From the center, measure a distance of 7 in all directions (up, down, left, right) to draw the circle.

Key Concepts

standard form of a circle

The standard form of a circle's equation is one of the most fundamental aspects when working with circles in the coordinate plane. This essential form is expressed mathematically as \[(x - h)^2 + (y - k)^2 = r^2\]where

• \((h, k)\) represents the center of the circle
• \(r\) denotes the radius.

By rewriting circle equations into this form, you can easily identify these vital components. Begin with your equation, rearrange it, and simplify until it resembles the standard form. Knowing the center and radius lets you visualize and graph the circle accurately, which is crucial for analysis and application of geometric properties.

completing the square

Completing the square is a technique used to manipulate a quadratic equation into a form that reveals more information about a geometric shape, like a circle. In the case of circles, it's used to convert a general quadratic equation into the standard form. Here's how it works:

For a quadratic expression such as \[x^2 + 2x\]you take half of the coefficient of \(x\), which is 2. You then square this result to get 1. This leads to the squared trinomial \[(x + 1)^2\].

For the \(y\) variable, perform a similar process. With \[y^2 + 12y\]take half of 12 to get 6, square it to acquire 36, forming \[(y + 6)^2\].Through these steps:

• Each squared term provides a transformation that makes graphing simple
• Helps in isolating the specific circle attributes from a more general quadratic equation.

This allows for quick visualization of the circle's position and size on the graph.

graphing circles

Graphing circles becomes intuitive once the equation is in standard form. Start by plotting the center point \((h, k)\) on the coordinate plane. For example, in our case, the center is at \((-1, -6)\).

From the center, use the radius \(r\), which you have determined from the equation. In our example, \(r = 7\). You will move outward from the center in all directions:

• Up
• Down
• Left
• Right

Mark these points at a distance equal to the radius and then smoothly connect them in a circular shape. Remember:

• The larger the radius, the bigger the circle.
• The position of the center shifts the circle around the plane accordingly.

This graphical representation of a circle helps you understand its properties dynamically and how it interacts with other geometric figures and points.