Problem 9 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=9 ; \text { down } 1 $$

Step-by-Step Solution

Answer

The equation of the circle after being translated one unit down is \(x^{2}+(y+1)^{2}=9\).

Step 1: Identify the Current Position of the Circle

The equation of the circle as given is \(x^{2}+y^{2}=9\). The circle is currently at position (0,0) because the equation can be written in the form \((x-h)^{2}+(y-k)^{2}=r^{2}\), where (h,k) is the center and r is the radius of the circle. In this case, h=0, k=0, and r=3.

Step 2: Apply the Translation

A downward translation of the circle requires subtracting the value of the translation (in this case, 1) from the y-coordinate in the equation. The new equation becomes \(x^{2}+(y+1)^{2}=9\). The center of the circle is now at (0,-1).

Step 3: Confirm the Changes

Confirm the transformation has taken place by comparing the two equations. The initial equation was \(x^{2}+y^{2}=9\), which represents a circle centered at (0,0), and the final equation is \(x^{2}+(y+1)^{2}=9\), which represents a circle centered at (0,-1). Therefore the circle has been translated one unit down as expected.

Key Concepts

Equation of a Circle

The equation of a circle is a foundational concept in geometry and algebra. It defines the set of all points that are equidistant from a central fixed point, known as the center. This equation can be represented in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, the variables \(h\) and \(k\) denote the coordinates of the circle's center, while \(r\) stands for the radius. When there is no shift along the x or y axes, the equation simplifies to \(x^2 + y^2 = r^2\), indicating a circle centered at the origin \((0,0)\).
Understanding how the equation is constructed helps in recognizing how different components affect the circle's position and size on a coordinate plane. By manipulating the equation, such as changing \(h\) or \(k\), we can shift the circle along the x and y axes, which is important for transformations.

Coordinate Transformation

Coordinate transformation involves altering the position of objects within a coordinate plane. This is done by adjusting the coordinates in the equation representing the object, affecting their place relative to other points on the plane. When discussing circles, this transformation is achieved by tweaking the terms \((x-h)\) and \((y-k)\), thereby moving the center.
To transform coordinates, we must understand the type of transformation required - whether it is a shift to the left, right, upward, or downward. Different types of translations can be applied:

• Horizontal Translations: Changes to the \(x\)-coordinate affect left-right movements.
• Vertical Translations: Adjustments to the \(y\)-coordinate result in up-down shifts.

For example, subtracting a unit from \(k\) means moving the circle downward by that unit, which directly affects the coordinate of the center.

Downward Shift

A downward shift is a type of vertical translation where an object is moved lower along the y-axis of a coordinate plane. For circles, this operation necessitates altering the \((y-k)\) term in the equation.
Consider the initial circle equation \(x^2 + y^2 = 9\). When applying a downward shift of 1 unit, the y-coordinate of the center changes, making \((y-k)\) become \((y+1)\). As a result, the circle is now centered at \((0,-1)\) rather than at the origin. This operation keeps the radius unchanged but alters the circle's position on the plane.
Understanding downward shifts is crucial because it demonstrates how changes to one part of an equation can move the entire figure, thereby influencing spatial relationships.