Problem 12 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=81 ; \text { left } 1 \text { and } \operatorname{up} 3 $$

Step-by-Step Solution

Answer

The equation of the translated circle is \((x + 1)^2 + (y - 3)^2 = 81\).

Step 1: Knowing the Original Equation

The original equation of the circle is \(x^2 + y^2 = 81\). This circle is centered at the origin (0,0) and has a radius of 9.

Step 2: Finding the New Coordinates for the Center

The center is to be shifted one unit to the left and three units upward. This means the new center will be at the coordinates \((-1, 3)\)

Step 3: Writing the Equation for the Translated Circle

Substituting \(-1\) for \(h\) and \(3\) for \(k\) in the general equation for a circle, the equation becomes \((x - (-1))^2 + (y - 3)^2 = 81\), or equivalently, \((x + 1)^2 + (y - 3)^2 = 81\)

Key Concepts

Circle Equation

The equation of a circle is a fundamental concept in geometry. It helps us understand the shape and position of a circle in a coordinate plane. The general form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Here:

• \( (h, k) \) denotes the center of the circle.
• \( r \) is the radius of the circle.

The expression \((x-h)^2 + (y-k)^2\) measures the squared distance from any point \((x, y)\) on the circle to its center. When this distance equals \( r^2\), the point lies exactly on the circumference of the circle.
In the original exercise, the given equation \(x^2 + y^2 = 81\) represents a circle centered at the origin, (0,0), with a radius of 9. This is because \(81\) is the square of the radius, meaning \(r = 9\).

Translations in Geometry

Translations are movements that shift a shape within a plane without altering its size or orientation. In essence, they are just a slide from one position to another. In mathematics, translations are precisely described using coordinate shifts.
Whenever you need to translate a geometric figure like a circle, you change the coordinates of its center. For the circle equation, this affects the \((h, k)\) part.

• If you move the circle to the left, you subtract from the x-coordinate.
• Moving it up means you add to the y-coordinate.

In the problem exercise, the translation involves a shift of the circle one unit to the left and three units upward. This changes the center from \((0, 0)\) to \((-1, 3)\). Translating it is like gently nudging the circle without stretching or rotating it, maintaining its perfect round shape.

Coordinate System

The coordinate system is a framework used to pinpoint locations on a flat plane. It includes two perpendicular axes: the X-axis (horizontal) and Y-axis (vertical), intersecting at the origin \((0, 0)\).
Each point on this plane is described with coordinates \((x, y)\), where:

• \(x\) indicates its horizontal position from the origin.
• \(y\) specifies its vertical position.

When plotting a circle, understanding this system is crucial because it dictates how we describe the circle's location and dimensions. The center's coordinates \((h, k)\) convey its precise position in this grid, while the radius \(r\) shapes its size.
In our translated circle example, we move from the origin to \((-1, 3)\) using the fundamentals of this coordinate system to place the newly positioned circle.

Graphing Circles

Graphing circles on a coordinate plane is a visual way to understand their position and size. Here’s a quick guide on how you might graph a circle:

• Identify the center, \((h, k)\), and mark it on the plane.
• From the center, measure out the radius \(r\) in all directions to form the circle's boundary.

The circle equation \((x-h)^2 + (y-k)^2 = r^2\) serves as the blueprint for drawing. Every point that satisfies this equation lies on the circle.
When graphing circles, the circle moves but retains its size if you perform a translation. For instance, after translating the circle in our exercise to \((-1, 3)\), you would plot the new center and keep the radius the same (9 units). This keeps the integrity of the circle intact while shifting its position on the plane. Graphing helps bring these mathematical concepts to life, allowing us to visualize and verify solutions.