Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x+3)^{2}+(y-1)^{2}=1$$

Step-by-Step Solution

Answer

Plot the circle with center \((-3, 1)\) and radius 1.

Step 1: Identify the Type of Equation

The given equation is \((x+3)^2 + (y-1)^2 = 1\). This is the standard form of a circle, \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center and Radius

From the equation \((x+3)^2 + (y-1)^2 = 1\), we can extract that the center of the circle is \((-3, 1)\) and the radius \(r\) is \(\sqrt{1} = 1\).

Step 3: Plot the Center of the Circle

On a graph, plot the center of the circle at the point \((-3, 1)\). This will be the middle point from which the circle extends.

Step 4: Draw the Circle Using the Radius

From the center \((-3, 1)\), measure and draw a circle with a radius of 1 unit in all directions (up, down, left, right). This means the circle will reach the points \((-2, 1), (-4, 1), (-3, 2), \text{ and } (-3, 0)\).

Key Concepts

Equation of a Circle

The equation of a circle in a Cartesian coordinate system helps you understand its shape and location. A circle's equation is most commonly presented in the standard form: \((x-h)^2 + (y-k)^2 = r^2\).
This form highlights two crucial components: the circle's center point \((h, k)\) and its radius \(r\).
To recognize that an equation represents a circle, ensure both \(x\) and \(y\) terms are squared and added together.In our example, the equation is given as \((x+3)^2 + (y-1)^2 = 1\).
Comparing this with the standard form:

• The term \((x+3)^2\) can be rewritten as \((x-(-3))^2\), indicating a horizontal shift by -3.
• The term \((y-1)^2\) represents a vertical shift up by 1.

This tells us the circle has been moved from the origin to the point (-3, 1). Next, we'll explore what this center point signifies.

Center of a Circle

The center of a circle is a pivotal point on the graph representing the exact middle of the circle. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the coordinates \((h, k)\) denote this center.
This point helps you understand where the circle is located within the plane. It is essential as all points forming the circle are equidistant from this center.If we analyze our specific equation \((x+3)^2 + (y-1)^2 = 1\), the transformations reveal:

• \(h = -3\): a horizontal shift 3 units left.
• \(k = 1\): a vertical shift 1 unit upwards.

With the center at \((-3, 1)\), this coordinates every point on the circle to have the same distance from \((-3, 1)\). Identifying the center is your first step towards graphing the circle correctly.

Radius of a Circle

The radius of a circle defines how far the circle reaches from its center to any point on the circle's edge.
It is a fixed distance, denoted as \(r\) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\).To find the radius, consider the equation we are working with: \((x+3)^2 + (y-1)^2 = 1\). In this example, the value on the right, \(1\), equals \(r^2\).
By taking the square root, you conclude that the radius \(r\) is:

• \(r = \sqrt{1} = 1\)

This means each point of the circle is 1 unit away from the center \((-3, 1)\). Understanding the radius is fundamental as it tells you how large the circle will be on the graph. With this, you can plot points exactly 1 unit in all directions from the center, forming a perfect circle.