Problem 63 - Algebra 2

Question

The graph of each equation is translated 2 units left and 3 units down. Write each new equation. $$ (x+2)^{2}+(y-1)^{2}=5 $$

Step-by-Step Solution

Answer

The new equation after the graph of the equation \((x+2)^{2}+(y-1)^{2}=5\) is translated 2 units left and 3 units down is \(x^{2}+(y+2)^{2}=5\).

Step 1: Understand the Translations

In this step, it is needed to acknowledge that a translation of 2 units to the left means to subtract 2 from the x-coordinate in the equation. Similarly, a translation of the 3 units down means to add 3 to the y-coordinate.

Step 2: Apply the Horizontal Translation

Subtract 2 from every x-coordinates in the equation, which leads to the transformation of \(x + 2\) to \(x + 2 - 2\), or simply \(x\). So now the equation becomes \(x^{2}+(y-1)^{2}=5\).

Step 3: Apply the Vertical Translation

Add 3 to every y-coordinate in the equation, which leads to the transformation of \(y - 1\) to \(y - 1 + 3\), or simply \(y + 2\). Hence, the final transformed equation becomes \(x^{2}+(y+2)^{2}=5\).

Key Concepts

Coordinate Transformations

Coordinate transformations are essential changes applied to the equations of graphs to move them around the coordinate plane without altering their shape. These transformations help describe how a graph can shift, stretch, or rotate. In the context of translations, we focus on shifting the graph horizontally or vertically. Both types of shifts preserve the size and form of the graph but change its position.
Understanding these transformations allows us to manipulate equations and visualize how their graphs appear in different locations.

Horizontal Translation

A horizontal translation moves the graph left or right along the x-axis. It involves altering the x-coordinate values in the equation. If a graph shifts left, we subtract from the x-coordinate. If it moves right, we add to the x-coordinate.
For instance, translating a graph 2 units left means replacing every instance of the x-coordinate with \(x + 2 - 2\), simplifying it to \(x\). This adjustment shifts the entire graph without changing its vertical position.

• Left shift: Subtract from x-coordinate
• Right shift: Add to x-coordinate

This straightforward change enables us to relocate graphs in meaningful ways.

Vertical Translation

Vertical translation involves shifting a graph up or down along the y-axis. This transformation affects the y-coordinate values. Moving a graph down means adding to the y-coordinate, while moving it up requires subtraction.
For example, if a graph is moved 3 units down, as in the original problem, we adjust the y-coordinate from \(y - 1\) to \(y - 1 + 3\), resulting in \(y + 2\). The graph's horizontal position remains unchanged.

• Downward shift: Add to y-coordinate
• Upward shift: Subtract from y-coordinate

Vertical translations allow the graph to shift its height on the coordinate plane, facilitating better alignment with real-world scenarios.