Problem 51 - College Algebra Essentials

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. $$ (x+1)^{2}+y^{2}=25 $$

Step-by-Step Solution

Answer

The center of the circle described by the equation \((x+1)^2 + y^2 = 25\) is at point \((-1, 0)\) and its radius is \(5\). The domain and range from the graph are both \([-6, 4]\) and \([-5, 5]\) respectively.

Step 1: Identify the Center and Radius

Firstly, the equation of the circle provided is \((x+1)^2 + y^2 = 25\), and this is in the form of \((x-h)^2 + (y-k)^2 = r^2\). So, comparing this with the standard circle equation, it can be seen that \(h = -1\), \(k = 0\), and \(r^2 = 25\). Thus, the center of the circle is \((-1, 0)\) and the radius \(r\) is \(\sqrt{25} = 5\)

Step 2: Graph the Circle

Plot the center of the circle \((-1, 0)\) on a graph, then draw a circle around the center point with a radius of \(5\). Make sure the circle touches the \(y\)-axis at points \((0, 5)\) and \((0, -5)\), and extends as far as \(x = -6\) and \(x = 4\).

Step 3: Identify the Domain and Range

From the graph, it can be observed that the values of \(x\) (the domain) extend from \(-6\) on the left of the graph to \(4\) on the right. Therefore, the domain is \([-6, 4]\). The values of \(y\) (the range), on the other hand, go from \(-5\) at the bottom of the graph to \(5\) at the top, so the range is \([-5, 5]\).

Key Concepts

Circle Equation

Understanding the equation of a circle is crucial in graphing circles in algebra. A standard circle equation comes in the form of \[ (x-h)^2 + (y-k)^2 = r^2 \], where \(h, k)\) is the center of the circle and \(r\) is its radius. In the given equation \[ (x+1)^2 + y^2 = 25 \], it's modified from the standard form, indicating that the circle has been shifted along the x-axis. Here, the \(h\) value is affected by the sign inside the brackets, so with \(x+1\), our \(h\) is \-1. There is no shift along the y-axis because the \(k\) value represented by \(y\) is 0. When solving this type of equation, it's important to recognize these shifts to correctly identify the circle's placement on the coordinate system.

Always remember to square the radius value on the right of the equation to complete it. In this problem, \(r^2\) is given as 25, which means that \(r\) is 5 after taking the square root. The beauty of the circle equation is that it encompasses all the points around the center that are equally distanced by the radius, forming a perfect circle.

Center and Radius of a Circle

The center and radius of a circle are possibly the most critical attributes when it comes to its graphical representation. From our circle equation \( (x+1)^2 + y^2 = 25 \), after comparing with the general form, we determine that the center of the circle \( (-1, 0) \) is located one unit to the left of the origin along the x-axis. The radius, a constant distance from the center of the circle to any point on its perimeter, is \( 5 \), as \( \sqrt{25} = 5 \).

When graphing, you first plot the center. Visualize radiating out in all directions the length of the radius to define the circle’s boundary. This visual assists in comprehending that no matter where you are on the circumference of the circle, you are always the same distance, its radius, from the center.

Domain and Range of a Relation

The domain and range represent the possible x and y values that a relation can have. For the circle equation \( (x+1)^2 + y^2 = 25 \), the domain is the set of x-values within the circle’s width and the range is the set of y-values within the circle’s height. After graphing the circle, we notice that the furthest left point is -6 and the furthest right is 4. Hence, the domain is \[ [-6, 4] \]. For the range, the lowest and highest points on the circle correspond to a y-value of -5 and 5, respectively, making the range \[ [-5, 5] \].

Understanding domain and range is like knowing the limits within which the circle lives on the coordinate plane. Every point of the circle exists within these limits, ensuring that when you graph, you’re capturing the full essence of the circle.

Plotting Circles on a Graph

Plotting a circle starts with marking the center. In our example, we plot the point \( (-1, 0) \) on the graph. To ensure accuracy when drawing the circle, pinpoint the locations where the radius meets the axes. These will be at \(x = -6\)\ and \(x = 4\) on the x-axis, and at \(y = -5\) and \(y = 5\) on the y-axis. Draw a smooth curve connecting these points, resulting in the circle’s edge touching these four points.

This process visualizes how the equation of a circle translates to a graphical representation. Plotting circles might require a little practice to make the curve smooth, but understanding the circle’s basic properties, such as the center and radius, makes the task significantly easier. Remember, the points where your drawn circle touches the axes are your guide to ensuring the graph accurately represents the circle's equation.