Problem 57 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$(x-2)^{2}+(y-2)^{2}=16$$

Step-by-Step Solution

Answer

The graph is a circle with center (2, 2) and radius 4.

Step 1: Identify the Type of Graph

The given equation \((x-2)^2 + (y-2)^2 = 16\) is in the standard form of a circle equation, which is \((x-h)^2 + (y-k)^2 = r^2\). Compare both equations to identify \(h\), \(k\), and \(r\).

Step 2: Determine the Center of the Circle

The circle equation \((x-h)^2 + (y-k)^2 = r^2\) has its center at \((h, k)\). By comparing it with \((x-2)^2 + (y-2)^2 = 16\), we determine that \(h = 2\) and \(k = 2\). Thus, the center of the circle is \((2, 2)\).

Step 3: Calculate the Radius of the Circle

In the circle equation \((x-h)^2 + (y-k)^2 = r^2\), \(r\) is the square root of the constant term on the right side of the equation. Thus, \(r = \sqrt{16} = 4\). The radius of the circle is \(4\).

Step 4: Sketch the Graph

With the center at \((2, 2)\) and a radius of \(4\), plot the center on a coordinate plane. From the center, measure out 4 units in all directions (up, down, left, and right) to sketch the circle. Connect these points smoothly to form the circle.

Key Concepts

Center of a Circle

The center of a circle is a key feature that determines its position on a coordinate plane. In the standard form of a circle's equation, \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by the coordinates \((h, k)\).
To find the center, identify the constants subtracted from x and y within the equation. For instance, if the equation is \((x-2)^2 + (y-2)^2 = 16\), compare it to the standard form:

• The first part, \((x-2)^2\), implies that \(h = 2\).
• The second part, \((y-2)^2\), shows that \(k = 2\).

Thus, the center of this circle is located at \((2, 2)\). Recognizing this, you can confidently locate where the circle is situated in relation to the axes.
Understanding how to determine the center from the equation will help you accurately graph circles and solve related mathematical problems.

Radius of a Circle

The radius of a circle represents the distance from the center to any point on the circle's boundary. This measurement is crucial when plotting or analyzing circles. It’s helpful to understand that in the equation \((x-h)^2 + (y-k)^2 = r^2\), the radius is denoted by \(r\).
To find the radius, take the square root of the number on the right side of the equation. For example, with the equation \((x-2)^2 + (y-2)^2 = 16\):

• This tells us that \(r^2 = 16\).
• Taking the square root, we find \(r = \sqrt{16} = 4\).

Thus, the radius of the circle is \(4\). Knowing the radius allows you to measure the extent of the circle from its center point outwards.
Having a clear grasp of how to derive the radius from the equation can improve both your understanding and your ability to visualize the circle in question.

Graphing Circles

Graphing circles on a coordinate plane involves understanding both the center and the radius. Once you know these two components, drawing the circle becomes much simpler.
The process starts by marking the center of the circle on the graph. For instance, if your center is \((2, 2)\), place a point on these coordinates.
Next, use the radius for your circle's size. A radius of \(4\) means measuring \(4\) units from the center point in all main directions: up, down, left, and right. You now have a guide for the circle’s boundary.

• From the center, mark points \(4\) units away in each of these directions.
• Carefully connect these boundary points in a smooth, rounded shape to form the complete circle.

Accurately graphing a circle requires precision, but once the center and radius are identified, the task is straightforward. This skill is valuable for visualizing and working with circle equations in mathematics.