Problem 11 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=25 ; \text { right } 2 \text { and down } 4 $$

Step-by-Step Solution

Answer

The equation of the circle after translation right by 2 units and down by 4 units is \((x-2)^2 + (y+4)^2 = 25\).

Step 1: Identify the circle equation

The equation of the original circle is given as \(x^{2}+y^{2}=25\). This, in standard form, is equivalent to \((x-0)^2 + (y-0)^2 = 5^2\), meaning the circle has its center at origin (0,0) and radius 5.

Step 2: Calculate the new center after translation

The circle is to be translated right 2 units and down 4 units. This means h (x-coordinate of the center) will increase by 2 units and k (y-coordinate of the center) will decrease by 4 units. The new center will be at (0+2, 0-4) which is (2, -4).

Step 3: Write the equation for the translated circle

Referring to the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), substitute the new center coordinates (h = 2, k = -4) and the original radius (r = 5). The new equation will be \((x-2)^2 + (y--4)^2 = 5^2\), which simplifies to \((x-2)^2 + (y+4)^2 = 25\). This is the equation of the circle after the translation.

Key Concepts

Coordinate Translation

When we talk about coordinate translation, we are essentially moving the entire graph of a shape to a new position on the coordinate plane. This movement does not alter the shape's size or orientation. It only changes its location. To perform a coordinate translation:

• Determine how many units you need to move the shape horizontally. Shifting a shape to the right means adding a specific number to the x-coordinate of every point on the shape. Shifting it to the left involves subtracting from the x-coordinate.
• Determine how many units to move the shape vertically. Moving a shape up means adding to the y-coordinate, while moving it down means subtracting from the y-coordinate.

In our exercise, the initial circle at the origin was moved right by 2 units and down by 4 units. This translation gave us the new center point of (2, -4). All points on the circle adjust in the same manner, ensuring the circle maintains its original shape and size.

Standard Form Equation

In geometry, circles are often expressed with a standard form equation. This helps in easily identifying key characteristics of the circle like its center and radius. The standard form equation for a circle is written as:\[ (x - h)^2 + (y - k)^2 = r^2 \]Where:

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

For example, in the given problem, the original circle had the equation \( x^2 + y^2 = 25 \). This can be rewritten in the standard form as \( (x-0)^2 + (y-0)^2 = 5^2 \), indicating a circle centered at the origin (0,0) with a radius of 5. Converting circle equations into their standard form can make it easier to perform transformations and understand the circle's properties.

Circle Transformation

Circle transformation refers to when a circle undergoes a change in its position, form, or size. In this context, we are focused on moving a circle around the coordinate plane, which is called translation. Let's break down how we transform a circle:

• Upon translation, the radius of a circle remains unchanged. This is because translation only shifts the circle's position without distorting its shape.
• To find the new equation of a translated circle, substitute the new center's coordinates into the standard form equation. For example, our exercise showed us the shift resulted in the new equation \((x-2)^2 + (y+4)^2 = 25\).
• Remember, the sign in the equation will be opposite to the direction of translation. If moved right, the value in the equation will decrease, and moving down will make the coefficient of the y-term positive.

Understanding how transformations affect a circle helps in visualizing and graphically presenting their geometric nature more vividly.