Problem 23 - College Algebra with Modeling and Visualization

Question

Write an equation that shifts the given circle in the specified manner. State the center and radius of the translated circle. \(x^{2}+y^{2}=4 ;\) right 3 units, downward 4 units

Step-by-Step Solution

Answer

The translated circle is centered at (3, -4) with radius 2.

Step 1: Identify Original Equation Parameters

The given equation of the circle is \(x^2 + y^2 = 4\). This represents a circle centered at the origin \((0, 0)\), with a radius of 2, because 4 is the square of the radius, i.e., \(r^2 = 4\).

Step 2: Determine New Center after Translation

To translate the circle, move it right by 3 units, which adds 3 to the x-coordinate, and downward by 4 units, which subtracts 4 from the y-coordinate. Therefore, the new center is \((0+3, 0-4) = (3, -4)\).

Step 3: Write the Equation of the Translated Circle

The general form of a circle's equation with center \((h, k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). Substituting the new center \((3, -4)\) and radius \(r=2\), the equation becomes \((x-3)^2 + (y+4)^2 = 4\).

Step 4: Summary of Translated Circle

The new circle is centered at \((3, -4)\) with a radius of 2. The equation \((x-3)^2 + (y+4)^2 = 4\) represents this translated circle.

Key Concepts

Circle Translation

When you hear the term **circle translation**, think about shifting a circle from one location on the coordinate plane to another. This movement is all about relocating the center of the circle without changing its size. The original position of the circle is given by its equation. In this exercise, we started with the equation \(x^2 + y^2 = 4\), which suggests that the circle is initially centered at the origin \(0, 0\). To translate a circle, focus on the changes to the x and y coordinates of the center.

In our example, translating the circle means:

• Moving "right 3 units": This increases the x-coordinate of the center by 3.
• Moving "downward 4 units": This decreases the y-coordinate by 4.

After this translation, the new center becomes \(3, -4\). Importantly, the circle's radius remains unchanged during translation because we are only relocating its position on the grid.

Center of a Circle

Understanding the concept of the **center of a circle** is crucial when dealing with circles on a graph. The center is essentially the point equidistant from all points on the circle. For an equation in the form \((x - h)^2 + (y - k)^2 = r^2\), the center is the point \((h, k)\).

In the original circle equation \(x^2 + y^2 = 4\), there are no expressions in parentheses subtracted by a number, which means, it's centered at \(0, 0\).

After translating the circle, you adjust the equation to reflect the new center. In our example, after moving right and down, our new center becomes \(3, -4\). This change is incorporated into the translated equation \((x - 3)^2 + (y + 4)^2 = 4\). By understanding how to find and modify the center of a circle, you can easily manipulate its position on the graph.

Radius of a Circle

Learning how to determine and use the **radius of a circle** helps you better understand a circle's geometry. The radius is a constant distance from the center to any point on the circle. In an equation, the radius is indicated by \(r\) in the equation format \((x - h)^2 + (y - k)^2 = r^2\).

For our given circle, \(x^2 + y^2 = 4\), the value next to the equals sign is 4, which represents \(r^2\). Thus, the radius \(r\) is \(\sqrt{4} = 2\). It's important to note that the act of translating the circle (shifting its center) doesn't affect the radius.

In our solution, even after we found the new center, the radius remains 2, and our translated circle has the equation \((x - 3)^2 + (y + 4)^2 = 4\). Knowing the radius allows you to shape the exact size and measure perfect circles on a graph.