Problem 46 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$(x+1)^{2}+(y-4)^{2}=25$$

Step-by-Step Solution

Answer

The circle has a center at (-1,4) and has a radius of 5. Its domain is [-6,4] and its range is [-1,9].

Step 1: Find the center and the radius

The equation \((x+1)^{2}+(y-4)^{2}=25\) is in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, we see that h=-1 and k=4. Therefore, the center of the circle is at (-1,4). The radius can be found by taking the square root of 25, which is 5.

Step 2: Graph the circle

Plot the center (-1,4) on your graph first. From this point, draw a circle with radius 5. Measure 5 units to the left, right, up, and down from the center, and connect these points to graph your circle.

Step 3: Identify the domain and range

The domain (values of x) are from the center, -1, plus and minus the radius. Therefore, the domain is from -6 to 4 inclusive. The range (values of y) is from the y-coordinate of the center, 4, plus and minus the radius. The range is from -1 to 9 inclusive.

Key Concepts

Center and Radius of a Circle

Understanding the center and radius of a circle is essential in graphing and analyzing it. Given the equation \(x+1)^2 + (y-4)^2 = 25\), let's break down its components. In this form, known as the standard form equation of a circle, the values \(h\) and \(k\) represent the x and y coordinates of the center, respectively. Here, \(h = -1\) and \(k = 4\), telling us the center is at the point (-1, 4).

To find the radius \(r\), observe the constant term on the equation's right side, which is the square of the radius. Taking its square root gives us \(r = 5\). The radius is a crucial component as it sets the size of the circle and helps determine the domain and range, affecting how we plot the circle on a coordinate plane.

Graphing Circles

Graphing circles can be a simple process once you have identified the circle's center and radius. First, place a point on the graph at the center, which in our case is at (-1, 4). Consider this your anchor point. Building from the center, use the radius to measure outwards in all directions—left, right, above, and below. For our radius of 5, you'll count five units from the center to these points. By connecting these points and ensuring the curve is even, you create the perimeter of the circle. Practice drawing smooth, round edges to represent the circle accurately. Making sure to draw the circle symmetrically ensures a clear understanding of its graphical representation.

Domain and Range of a Relation

The domain and range of a relation define the set of possible input (x-values) and output (y-values) for the relation—in this case, our circle.

Domain

For a circle, the domain consists of all x-values within the distance of the radius from the circle's center horizontally. For our circle equation, the center is at x = -1, and the radius is 5. Consequently, the domain extents from \(x = -1 - 5\) to \(x = -1 + 5\), giving us the domain [-6, 4].

Range

Similarly, for the range, we consider the y-values. Starting from the center's y-coordinate, which is 4, we can move up and down by the radius of 5. Thus, the range goes from \(y = 4 - 5\) to \(y = 4 + 5\), resulting in the range [-1, 9]. The domain and range are crucial for understanding how extensive the circle's presence is on the coordinate plane.

Standard Form Equation of a Circle

The standard form equation of a circle plays a vital role in geometry as it succinctly conveys the properties of a circle. Such an equation is represented as \( (x-h)^2 + (y-k)^2 = r^2 \)). In this equation, the variables \((h,k)\) denote the circle's center, and \(r\) signifies the radius. The beauty of this form lies in its directness; it provides the information needed to draw the circle without further computation. Our example equation \( (x+1)^2 + (y-4)^2 = 25 \) translates to a circle with a center at (-1, 4) and a radius of 5. Clearly understanding this standard form helps students and mathematicians effortlessly identify the fundamental attributes of any circle.