Problem 37 - A Graphical Approach to College Algebra

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

The circle's center is (-3, -2) with a radius of 6. Domain: [-9, 3]; Range: [-8, 4].

Step 1: Identify the general form of a circle equation

The general form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) represent the center and \(r\) is the radius. Our equation is \((x+3)^2 + (y+2)^2 = 36\).

Step 2: Determine the center and radius

By comparing the given equation \((x+3)^2 + (y+2)^2 = 36\) with the general form \((x-h)^2 + (y-k)^2 = r^2\), we identify that \(h = -3\), \(k = -2\), and \(r^2 = 36\). Thus, the center is \((-3, -2)\) and the radius is \(r = \sqrt{36} = 6\).

Step 3: Graph the circle

To graph the circle, plot the center at \((-3, -2)\) on the coordinate plane. From this center, measure a distance of 6 units in all directions (up, down, left, and right) to draw the circle. The diameter is 12 units.

Step 4: Determine the domain of the circle

The domain of a circle is the set of all possible \(x\)-values that the circle covers. Since the circle's center is at \(x = -3\) and the radius is 6, the circle spans from \(-3 - 6\) to \(-3 + 6\) on the x-axis. Therefore, the domain is \([-9, 3]\).

Step 5: Determine the range of the circle

The range of a circle is the set of all possible \(y\)-values that the circle covers. Since the circle's center is at \(y = -2\) and the radius is 6, the circle spans from \(-2 - 6\) to \(-2 + 6\) on the y-axis. Therefore, the range is \([-8, 4]\).

Key Concepts

Domain and Range

In mathematics, the domain and range describe the set of possible values for the functions of a shape or figure. For circles, which are represented by an equation like \((x-h)^2 + (y-k)^2 = r^2\):

• Domain: This is the set of all possible \( x \)-values covered by the circle. For a circle with center \((-3, -2)\) and radius 6, the domain is calculated from the center along the x-axis. The circle reaches from \(x = -3 - 6 = -9\) to \(x = -3 + 6 = 3\). Hence, the domain is \([-9, 3]\).
• Range: This is the set of all possible \( y \)-values covered by the circle. Similar to domain, for the given circle, the range spans from \(y = -2 - 6 = -8\) to \(y = -2 + 6 = 4\). Therefore, the range is \([-8, 4]\).

Understanding the domain and range is vital for graphing a circle as it shows the horizontal and vertical extents on the coordinate plane.

Graphing Circles

To graph a circle, begin by identifying its equation in the form \((x-h)^2 + (y-k)^2 = r^2\). This gives you all the necessary information: the center \((h, k)\) and the radius \(r\). For the equation \((x+3)^2 + (y+2)^2 = 36\), it's clear that the center is \((-3, -2)\) and the radius \(r\) is 6.

• Step-by-Step Graphing:
  • Locate the Center: Plot the point \((-3, -2)\) on a coordinate plane.
  • Use the Radius: From the center, move 6 units in every direction (up, down, left, and right) to help sketch the circle.
  • Sketch the Circle: Connect these boundary points to form the circular shape.

Graphing circles is a straightforward process once you break down its components and follow the steps carefully.

Center of a Circle

The center of a circle is a crucial point that defines its position in space. It's represented by the coordinates \((h, k)\) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\). For the circle equation \((x+3)^2 + (y+2)^2 = 36\), we determine the center as \((-3, -2)\).

• Understanding the Center: The values \(h\) and \(k\) are found by rewriting the terms \((x+3)^2\) and \((y+2)^2\) as \((x - (-3))^2\) and \((y - (-2))^2\), revealing the negative shifts from the origin.
• Role of the Center: It acts as a pivot for drawing the circle, ensuring it maintains a consistent radius. It's essentially the middle point that the whole circle revolves around.

The center makes it simple to find the domain and range and is the anchoring point for the circle's shape.

Radius of a Circle

The radius is the distance from the center of the circle to any point on its perimeter. Represented by \(r\) in the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), it plays a critical role in defining the size of the circle. From the given equation \((x+3)^2 + (y+2)^2 = 36\), we identify the radius \(r = \sqrt{36} = 6\).

• Calculating the Radius: The equation equates to \(r^2\), so solve for \(r\) by computing the square root of that constant on the right side of the equation. In our case, it's \(36\), so \(r = 6\).
• Importance of the Radius: This distance determines how far each point on the circle's edge spreads from its center. A larger radius means a larger circle.

Understanding the radius is key to both graphing and comprehending the circle's basic properties.