Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Graph each circle by hand if possible. Give the domain and range. $$(x+3)^{2}+(y+2)^{2}=36$$

Step-by-Step Solution

Answer

Center: (-3, -2), Radius: 6, Domain: [-9, 3], Range: [-8, 4].

Step 1: Identify the Circle's Equation Form

The given equation of the circle is \((x + 3)^2 + (y + 2)^2 = 36\). This is in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\) where \( (h, k) \) is the center and \( r \) is the radius.

Step 2: Determine the Center of the Circle

To find the center, compare the given equation to the standard form. The terms \((x+3)^2\) and \((y+2)^2\) imply that \( h = -3 \) and \( k = -2 \). Thus, the center of the circle is \((-3, -2)\).

Step 3: Find the Radius of the Circle

The equation has \( 36 \) on the right side, which represents \( r^2 \). Therefore, the radius \( r \) is the square root of 36, which is 6.

Step 4: Graph the Circle

To graph the circle, plot the center at \((-3, -2)\) on the Cartesian plane. Then, from the center, measure out a distance of 6 units in all directions (up, down, left, right, and diagonally) to sketch the circle maintaining a consistent distance (radius) from the center.

Step 5: Determine the Domain of the Circle

The domain of a circle consists of all the possible x-values it can have. Since the center is at \( x = -3 \) and the radius is 6, the x-values range from \(-3 - 6 = -9\) to \(-3 + 6 = 3\). Thus, the domain is \([-9, 3]\).

Step 6: Determine the Range of the Circle

The range of the circle consists of all possible y-values it can have. The center is at \( y = -2 \) with a radius of 6, providing y-values from \(-2 - 6 = -8\) to \(-2 + 6 = 4\). Hence, the range is \([-8, 4]\).

Key Concepts

Domain and Range

The concepts of domain and range are fundamental when graphing any function or shape, including a circle.
For a circle centered at a point \((h, k)\) with radius \(r\), the domain represents all possible \(x\)-values that the circle can have. Since the x-values span from a distance equal to the radius on either side of the center, we calculate the domain as follows:

• Start from the x-coordinate of the center, add and subtract the radius.
• Here, the center is at \(-3\) and the radius is \(6\).
• The domain then ranges from \(-3 - 6 = -9\) to \(-3 + 6 = 3\).

Therefore, the domain of this circle is represented as \([-9, 3]\).
The range follows a similar pattern but focuses on the y-values. It tells us the possible vertical span of the circle:

• Begin from the y-coordinate of the center, modify it by adding and subtracting the radius.
• In this case, the center has a y-value of \(-2\) and the radius is \(6\).
• Thus, the range extends from \(-2 - 6 = -8\) to \(-2 + 6 = 4\).

As a result, the range of the circle is \([-8, 4]\).
By understanding domain and range, you can determine the full extent of the circle on a graph, horizontally and vertically.

Center of the Circle

The center of a circle is a crucial piece of information when graphing or analyzing the circle's properties.
In a circle's standard equation form, \( (x - h)^2 + (y - k)^2 = r^2\), the center is represented by the coordinates \( (h, k) \).
It acts as a fixed point from which all points on the circumference are equidistant.
For the given equation \((x + 3)^2 + (y + 2)^2 = 36\):

• The term \(x + 3\) implies that the x-coordinate of the center (\