Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Graph each circle by hand if possible. Give the domain and range. $$(x+2)^{2}+(y-5)^{2}=16$$

Step-by-Step Solution

Answer

Center: (-2, 5); Radius: 4; Domain: [-6, 2]; Range: [1, 9]

Step 1: Identify Center and Radius

The equation of a circle is generally given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. For the given circle \((x + 2)^2 + (y - 5)^2 = 16\), we can rewrite it in terms of the standard equation: \((x - (-2))^2 + (y - 5)^2 = 4^2\). This reveals that the center \((h, k)\) is \((-2, 5)\) and the radius \(r\) is 4.

Step 2: Graph the Circle

To graph the circle, start by plotting the center \((-2, 5)\) on the coordinate plane. From the center, measure a distance of 4 units in all directions (up, down, left, and right) to accommodate the radius. Mark these points, then draw a smooth curve connecting them to form the circle.

Step 3: Determine the Domain

The domain of a circle involves the set of all possible \(x\)-values. In this case, since the leftmost point of the circle is at \((-2 - 4, 5)\) or \((-6, 5)\) and the rightmost point is at \((-2 + 4, 5)\) or \((2, 5)\), the domain is \([-6, 2]\).

Step 4: Determine the Range

The range of a circle involves the set of all possible \(y\)-values. Here, the lowest point is at \((-2, 5 - 4)\) or \((-2, 1)\) and the highest point is at \((-2, 5 + 4)\) or \((-2, 9)\), making the range \([1, 9]\).

Key Concepts

Domain and Range

The domain and range of a circle define all the possible values that the circle's equation can hold for the variables \(x\) and \(y\).
In simpler terms, the domain is the complete set of \(x\)-values (horizontal coordinates), while the range is the entire set of \(y\)-values (vertical coordinates).

Let's explore this using our given circle equation:

• Domain: Since the circle is centered at \((-2, 5)\) and has a radius of 4, it stretches horizontally from \(x = -2 - 4\) (leftmost point) to \(x = -2 + 4\) (rightmost point), giving us a domain of \([-6, 2]\).
• Range: Vertically, the circle extends from \(y = 5 - 4\) (lowest point) to \(y = 5 + 4\) (highest point), providing a range of \([1, 9]\).

Understanding domain and range is crucial for graphing because it helps identify the extent of the circle on a coordinate plane.

Equation of a Circle

The equation of a circle is a fundamental concept in mathematics that allows us to represent a circle in a coordinate plane using a simple algebraic formula.
This standard equation is given by \[(x-h)^2 + (y-k)^2 = r^2\]where:

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

For the given circle \[(x+2)^{2}+(y-5)^{2}=16\],we translate this into the standard format as follows:

• Rewrite \((x+2)^2\) as \((x - (-2))^2\)\ and \((y-5)^2\){} remains unchanged.
• This shows the circle's center \((h, k) = (-2, 5)\)\ and the radius squared is 16, so \(r = 4\).

This equation helps in visually plotting the circle by defining its position (center) and size (radius). It is the key to interpreting the relationships between the circle's algebraic and geometric properties.

Center and Radius

The center and radius are two critical elements of a circle that define its unique position and scale on the coordinate plane.
Let's break down what each means and how we identify them in our equation.

The center of the circle is the point inside the circle that is equidistant from all points on the boundary.

• In the equation \((x-h)^2 + (y-k)^2 = r^2\), the values \(h\) and \(k\) pinpoint this center.
• For our example, \((x + 2)^2 + (y - 5)^2 = 16\), the center is located at \((-2, 5)\).

The radius is the distance from the center to any point on the circle itself.

• In the same equation, \(r^2\) is the value you have on the right side, indicating the radius squared.
• So, for a radius squared of 16, the radius is \(r = 4\).

Knowing the center and radius not only helps plot a perfect circle but also assists in understanding the spatial properties of the circle within the coordinate plane.