Problem 10 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The equation representing the given circle translated one unit to the left is \((x+1)^{2}+y^{2}=1\).

Step 1: Representation of Translation

For an equation, any horizontal translation to the left is represented by replacing 'x' in the equation with '(x + 1)'.

Step 2: Apply Translation

Applying this to the given equation \(x^{2}+y^{2}=1\), replace each 'x' with '(x + 1)'. The equation becomes \((x+1)^{2}+y^{2}=1\).

Key Concepts

Horizontal Translation of a Circle Equation

In geometry, when dealing with circle equations, a horizontal translation involves shifting the entire circle to the left or right. This is specifically focused on moving the circle along the x-axis without altering its size or shape. To understand how this works in mathematics, consider the circle equation given by the standard form \(x^2+y^2=1\). A **horizontal translation** to the left requires modifying each 'x' in the equation. You do this by replacing 'x' with \(x + h\), where 'h' is the number of units you want to translate the circle.
For example, if you wish to move the circle to the left by 1 unit, you replace each 'x' with \(x + 1\) in the equation. The new equation becomes \((x+1)^2 + y^2 = 1\). This translation retains the circle’s radius and center position along the y-axis while the center on the x-axis shifts accordingly.

Translation in Geometry

Translation in geometry refers to moving a shape or object in a two-dimensional plane without changing its orientation. Think of it as sliding the shape, so its position changes, but its look remains constant. This motion is characterized by a precise direction and a specific distance.

Key points to remember about translation:

• **Direction**: The movement can either be horizontal or vertical, or a combination of both.
• **Distance**: The number of units the shape is moved.
• **Result**: The size, shape, and orientation remain unchanged, which means translation is an isometry.

In this exercise, translation is only affecting the x-axis, indicating a purely horizontal shift. This simplification makes it easier to see how geometric translation affects equations directly.

Equation Transformation through Translation

Equation transformation occurs when you apply a translation to the terms of a geometric equation. This transformation allows us to see how changes in position impact the algebraic representation of geometric shapes.

When translating equations:

• Identify which axis the translation affects (in our example, it is the x-axis).
• Substitute the variable with its translated form (e.g., replacing 'x' with \(x + 1\) for a left move).
• Recalculate or simplify the new equation while keeping the properties of the geometric figure intact.

Through this kind of transformation, the given circle equation changes from \(x^2+y^2=1\) to \((x+1)^2 + y^2=1\). This highlights how a simple replacement can change the position of the shape on the plane, illustrating the power and elegance of equation transformation in geometry.