Problem 38 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, and ellipses. \((x-3)^{2}+(y+1)^{2}=25\)

Step-by-Step Solution

Answer

The equation is a circle centered at \((3, -1)\) with a radius of \(5\).

Step 1: Recognize the Equation Type

The given equation \((x-3)^{2}+(y+1)^{2}=25\) is in the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) are the coordinates of the center, and \(r\) is the radius of the circle.

Step 2: Identify Center and Radius

Compare the given equation \((x-3)^{2}+(y+1)^{2}=25\) with the standard form. We can identify \(h = 3\), \(k = -1\), and \(r^2 = 25\). Thus, \(r = \sqrt{25} = 5\). The center is \((3, -1)\) and the radius is \(5\).

Step 3: Graph the Circle

To graph the circle, plot the center at \((3, -1)\) on the coordinate plane. Then, using the radius of \(5\), draw a circle around the center that reaches \(5\) units away from it in all directions. This includes points such as \((8, -1)\), \((3, 4)\), \((-2, -1)\), and \((3, -6)\), all lying on the circle.

Key Concepts

Standard Form

The equation of a circle in its "Standard Form" is a crucial concept for understanding circles in coordinate geometry. The standard form of a circle's equation is written as

• \((x-h)^2 + (y-k)^2 = r^2\)

This equation highlights some essential components:

• \(h\) and \(k\) are the center's coordinates.
• \(r\) is the radius, squared to give \(r^2\).

The structure of this equation makes it easier to identify critical properties of a circle, such as its center and radius.
When you see an equation like \((x-3)^2+(y+1)^2=25\), it's already in standard form. From this, you can directly compare it with our blueprint to determine the circle’s features.
Identifying such equations helps one to then progress into graphing them efficiently.

Graphing Circles

"Graphing Circles" might sound tricky, but it's much simpler once you understand the fundamentals. To graph a circle represented by its equation in standard form, follow these steps:

• Identify the center \(h, k\).
• Determine the radius by taking the square root of \(r^2\).
• Plot the center on the coordinate plane.
• Draw a circle around the center with the determined radius.

Start by pinpointing the circle's center. For example, in the equation \((x-3)^2+(y+1)^2=25\), the center is at \((3, -1)\).
Next, move in a radius of 5 units outward from the center in all directions. This method means you'll cover all points that are 5 units away, thereby accurately portraying the circle on your graph.
Some practice by sketching different circle equations will significantly bolster your confidence in graphing them effortlessly.

Radius and Center Identification

Identifying the "Radius and Center" of a circle is a pivotal step in understanding and utilizing its equation.
The components \((h, k)\) in the equation \((x-h)^2 + (y-k)^2 = r^2\) allow you to spot the circle's center.To find the center from an equation like \((x-3)^{2}+(y+1)^{2}=25\), compare it directly to the standard form. You can see here that \(h = 3\) and \(k = -1\), so the center is \((3, -1)\).
The radius is another important item to ascertain. First, identify \(r^2\); for this equation, it's 25. Get the radius itself by taking the square root, which gives \(r = 5\). Collectively, knowing both the center and radius empowers you to graph a more precise circle and aids in understanding its positioning in relation to other geometrical figures. Being adept at these recognitions is central to mastering circle equations.