Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the equation of a circle with the given center and radius. Graph the circle. Center \((0,-3),\) radius \(\sqrt{7}\)

Step-by-Step Solution

Answer

The equation is \(x^2 + (y + 3)^2 = 7\). Graph the circle with center at (0, -3) and radius \(\sqrt{7}\).

Step 1: Identify Circle Formula

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

Step 2: Substitute Center into Formula

Plug the center coordinates \((0, -3)\) into the formula: \((x - 0)^2 + (y - (-3))^2 = r^2\). Simplifying, this becomes \(x^2 + (y + 3)^2 = r^2\).

Step 3: Substitute Radius into Formula

Substitute the radius \(\sqrt{7}\) into the equation: \(x^2 + (y + 3)^2 = (\sqrt{7})^2\). This simplifies to \(x^2 + (y + 3)^2 = 7\).

Step 4: Write Equation

The center-radius equation for the circle is \(x^2 + (y + 3)^2 = 7\).

Step 5: Graph the Circle

To graph the circle, plot the center at (0, -3), then draw a circle around it with a radius of \(\sqrt{7}\), which is approximately 2.65 units.

Key Concepts

Center-Radius Form

The center-radius form of a circle's equation is a very useful way to understand and describe circles in algebra. This equation is generally written as:

• \((x - h)^2 + (y - k)^2 = r^2\)

In this formula,

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius, which is the distance from the center of the circle to any point on its circumference.

This form is called the "center-radius form" because it highlights the circle's center and radius clearly. This makes it easy to quickly identify these properties when you encounter a circle equation.

Graphing Circles

Graphing a circle is straightforward once you have its center-radius form equation. Here are the key steps:

• Start by locating the circle's center on the graph. If the center is \((0, -3)\), you would go to \(0\) on the x-axis and \(-3\) on the y-axis.
• Next, use the radius to draw the circle. The radius is the constant distance from the center to any point on the circle. For a radius of\(\sqrt{7}\), which is around 2.65, measure this distance from the center in all directions.
• Drawing the circle means plotting points at this consistent radius all around the center and connecting them smoothly.

By following these steps, you'll end up with an accurate representation of the circle.

Radius Substitution

Replacing or substituting in the radius is a simple but crucial step when working with the center-radius form.Given a circle center \((0, -3)\) and a radius of \(\sqrt{7}\), substitute these into \((x - h)^2 + (y - k)^2 = r^2\).

• Substitute \(r\) with \(\sqrt{7}\). This requires you to square the radius to find \(r^2\).
• Therefore, \((\sqrt{7})^2 = 7\).

This substitution gives you the specific circle equation: \(x^2 + (y + 3)^2 = 7\), clearly providing the radius's role in forming the equation.

Circle Equation

The circle equation helps define the set of all points that make up the circle. Each point \((x, y)\) on the circle satisfies the equation:\(x^2 + (y + 3)^2 = 7\)This particular equation:

• Represents a circle with a center at \((0, -3)\)
• Has a radius of \(\sqrt{7}\), meaning any point on this circle is \(\sqrt{7}\) units away from the center.

Understanding this equation allows you to identify and analyze the properties and dimensions of the circle whenever you apply the center-radius form. This insight is essential for visualizing and solving problems that involve circles.