Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ (x+3)^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

The circle is centered at \((-3, 0)\) with radius 4.

Step 1: Identify the Circle Equation

The given equation of the circle is \((x+3)^2 + y^2 = 16\). This is in the standard form \((x-h)^2 + (y-k)^2 = r^2\) where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center of the Circle

From the equation \((x+3)^2 + y^2 = 16\), we can see that it can be rewritten in the form \((x-(-3))^2 + (y-0)^2 = 16\). Hence, the center of the circle is \((-3, 0)\).

Step 3: Find the Radius of the Circle

The right side of the equation \((x+3)^2 + y^2 = 16\) is \(16\), which is \(r^2\). Therefore, the radius \(r = \sqrt{16} = 4\).

Step 4: Sketch the Circle

To sketch the circle, plot the center at \((-3, 0)\) on a coordinate plane. Then, use the radius \(4\) to draw a circle around this center by marking points that are 4 units away from \((-3, 0)\) in all directions (left, right, up, and down).

Step 5: Identify Specific Points

The points on the circle are limits where each coordinate changes while keeping the distance of 4. Some points include: \((-3+4, 0) = (1, 0)\), \((-3-4, 0) = (-7, 0)\), \((-3, 0+4) = (-3, 4)\), and \((-3, 0-4) = (-3, -4)\). These points help guide the sketch.

Key Concepts

Circle Equation

The equation of a circle in a plane is a powerful tool in coordinate geometry. A standard circle equation looks like this: \((x-h)^2 + (y-k)^2 = r^2\). Here, \((x, y)\) represent the coordinates of any point on the circle. Meanwhile, \((h, k)\) are the coordinates of the circle's center, and \(r\) is the radius.

This form allows us to easily find the center and radius from the equation itself. For instance, the equation \((x+3)^2 + y^2 = 16\) can be directly compared with the standard form.

By identifying changes with \(x\) and \(y\), you can unravel the circle's center and its radius. This equation essentially captures all the points equidistant from a central point.

Center of the Circle

The center of a circle located on a coordinate plane is a fixed point from which every point on the circle is equidistant. In mathematical terms, this is represented in the circle's equation. Referring to the standard form \((x-h)^2 + (y-k)^2 = r^2\), the values of \(h\) and \(k\) help us pinpoint the circle's center.

In our example equation, \((x+3)^2 + y^2 = 16\), we rewrite it as \((x-(-3))^2 + (y-0)^2 = 16\).

This reveals that the center is at the coordinate \((-3, 0)\). Thus, at these coordinates, the center acts as the pivot for constructing the circle on a graph.

Radius of the Circle

The radius of a circle is the constant distance from its center to any point on its boundary. In the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the radius is represented by \(r\).

To find the radius from the equation, look at the number on the right. In our example, \((x+3)^2 + y^2 = 16\), the number \(16\) is equivalent to \(r^2\).

We derive the actual radius by computing \(\sqrt{16} = 4\). Thus, the circle's radius is 4 units. This length is crucial as it assists in sketching the circle accurately, keeping uniform distance from the center point.

Coordinate Plane

A coordinate plane is a fundamental concept in algebra and geometry, consisting of a two-dimensional surface where every point is uniquely defined by a pair of numbers, known as coordinates \((x, y)\). It features two intersecting lines: the horizontal one called the x-axis and the vertical one called the y-axis.

Using the coordinate plane, we can visualize the entire circle from the circle equation. For the equation \((x+3)^2 + y^2 = 16\), you start by marking the center \((-3, 0)\).

From the center, use the radius (4 units in our case) to measure outwards in all cardinal directions. This ensures that when you draw the curve, it remains at a constant distance from the center. Thus, the coordinate plane acts as a stage for graphing circles efficiently.