Problem 15 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=20 ; \text { left } 6 \text { and up } 1 $$

Step-by-Step Solution

Answer

The equation of the circle after translation is \((x+6)^{2}+(y-1)^{2}=20\).

Step 1: Translate Left

To translate the circle 6 units to the left, we substitute \(x+6\) for \(x\). So the equation becomes \((x+6)^{2}+y^{2}=20\).

Step 2: Translate Up

To translate the circle 1 unit up, we substitute \(y-1\) for \(y\). Now the translated equation becomes \((x+6)^{2}+(y-1)^{2}=20\).

Step 3: Final Equation

The final equation of the circle after being translated 6 units left and 1 unit up is \((x+6)^{2}+(y-1)^{2}=20\).

Key Concepts

Translations of Functions

Translations of functions are a fundamental concept in algebra that allow us to shift a graph along the coordinate plane without altering its shape. This is crucial when working with the equation of a circle, such as the one given in the original exercise. When we talk about translating a function, we mean moving it either horizontally or vertically on the graph:

• Horizontal translation shifts the graph left or right.
• Vertical translation moves the graph up or down.

In the context of a circle's equation, translating left or right involves adjusting the x-coordinate, while translating up or down involves modifying the y-coordinate. These translations do not affect the radius or the overall appearance of the circle on the graph; they simply reposition it.

Coordinate Shifts

Coordinate shifts refer to alterations made to the coordinates of the function's graph. These are expressed algebraically in the equation of the function.The circle's equation \[x^{2} + y^{2} = 20\]can be readjusted using coordinate shifts as specified in the exercise:

• To shift the circle left by 6 units, substitute \(x+6\) for \(x\). The equation becomes \((x+6)^{2} + y^{2} = 20\).
• To shift the circle up by 1 unit, substitute \(y-1\) for \(y\). The equation transitions to \((x+6)^{2} + (y-1)^{2} = 20\).

These coordinate shifts have real-world applications, such as repositioning objects or paths in graphic designs, ensuring that mathematical models remain accurate despite changes in point of reference.

Algebraic Transformations

Algebraic transformations are operations that alter an algebraic equation to represent shifts, stretches, or other changes in the graph of a function. Specifically, these transformations are applied to the variable terms in the equation.In our specific problem, we applied a transformation to translate the circle. It involved:

• Adding 6 inside the squared term for \(x\), resulting in \((x+6)^{2}\), to move it 6 units left.
• Subtracting 1 inside the squared term for \(y\), leading to \((y-1)^{2}\), to shift it 1 unit up.

These types of transformations are essential for customizing equations to fit specific criteria or conditions. They offer a simple way to explore how functions behave as their graphs change position or orientation.