Problem 36 - A Graphical Approach to College Algebra

Question

Graph each circle by hand if possible. Give the domain and range. $$(x-4)^{2}+(y-3)^{2}=25$$

Step-by-Step Solution

Answer

Center: (4, 3), Radius: 5; Domain: [-1, 9], Range: [-2, 8].

Step 1: Identify the Center and Radius

The standard form of a circle is \((x-h)^2 + (y-k)^2 = r^2\). Comparing this with the given equation \((x-4)^2 + (y-3)^2 = 25\), we identify the center of the circle as \((h, k) = (4, 3)\). The radius \(r\) is the square root of 25, which is \(5\).

Step 2: Graph the Circle

To graph the circle, start by plotting the center at the point \((4, 3)\) on a coordinate plane. Using the radius \(r = 5\), draw a circle around this center point, making sure the circle extends 5 units in all directions (up, down, left, right) from the center.

Step 3: Determine the Domain

The domain of a circle is the set of all possible x-values. Since the radius is 5 and the center is at \((4, 3)\), the circle extends from \(4 - 5 = -1\) to \(4 + 5 = 9\). Therefore, the domain is \([-1, 9]\).

Step 4: Determine the Range

The range of a circle is the set of all possible y-values. With a radius of 5 and center at \((4, 3)\), the circle extends from \(3 - 5 = -2\) to \(3 + 5 = 8\). Therefore, the range is \([-2, 8]\).

Key Concepts

Equation of a Circle

A circle is a perfect round shape in geometry, and its equation is often written in a standard form. The standard form equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\). This equation might look a bit complicated at first, but it becomes much clearer once you understand its components.

The variables \(h\) and \(k\) denote the coordinates of the center of the circle. Meanwhile, the term \(r\) represents the radius of the circle. The radius is the distance from the center to any point on the circle. In essence, the equation tells you everything you need to know about the size (through the radius \(r\)) and position (through the center \(h, k\)) of the circle.

As an example, consider the equation \((x-4)^2 + (y-3)^2 = 25\). It matches the standard form, and we can see by comparison that \((h, k) = (4, 3)\) and \(r^2 = 25\). Thus, the radius \(r\) is the square root of 25, which is 5. Once you understand these key parts of the equation, graphing and working with circles becomes a lot easier.

Center and Radius

The center and radius of a circle are fundamental characteristics that define a circle's position and size. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is represented by the point \((h, k)\). This is the fixed position around which the entire circle is symmetrically placed.

The center point is crucial when you need to graph a circle since it's the starting point from which you will measure the distance \(r\) to draw the circle. The radius \(r\) tells you how far from the center the circle extends in all directions. You find the radius by solving for \(r\) in \(r^2\) in the circle's equation.

For instance, with the equation \((x-4)^2 + (y-3)^2 = 25\), the center is \((4, 3)\) and the radius is 5 (since \(r^2 = 25\) and \(r = \sqrt{25}\)). The circle extends 5 units upward, downward, leftward, and rightward from this center point. Understanding how these two parts---center and radius---work helps in sketching and analyzing circles on the coordinate plane more efficiently.

Domain and Range

When graphing a circle, it's important to understand the concepts of domain and range, which refer to the set of possible x-values and y-values, respectively, for points on a circle.

The domain of a circle refers to every possible value \(x\) that can fit into the circle's equation. If the center is \((h, k)\) and the radius is \(r\), the circle stretches from \(h-r\) to \(h+r\) on the x-axis. This gives the domain as \([h-r, h+r]\).

To determine the range, which represents the possible y-values, you look at how far the circle extends vertically. From the circle's center \(k\), it ranges from \(k-r\) to \(k+r\) on the y-axis.

In our example of a circle with equation \((x-4)^2 + (y-3)^2 = 25\), the center is \((4, 3)\) with a radius of 5. The domain is calculated as \([4-5, 4+5]\), which simplifies to \([-1, 9]\). Similarly, the range is \([3-5, 3+5]\), giving \([-2, 8]\). This means every point along the circle falls within these x and y ranges. Understanding domain and range helps you graph and analyze circles within the limitations of a coordinate system.