Problem 13 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=100 ; \text { down } 5 $$

Step-by-Step Solution

Answer

The equation \(x^{2}+(y-5)^{2}=100\) is the result of translating the original equation \(x^{2}+y^{2}=100\) down by 5 units.

Step 1: Understanding the transformation

The given transformation is a downward translation by 5 units. The standard rule for a downward translation of a function \(f(x,y)\) is replacing \(y\) with \(y+k\), where \(k\) is the number of units the function is translated. Here, we are translating down, so \(k=-5\).

Step 2: Apply the transformation to the equation

To apply the transformation, replace \(y\) in the equation \(x^{2}+y^{2}=100\) with \(y-5\). This gives the equation \(x^{2}+(y-5)^{2}=100\).

Key Concepts

Translations in Coordinate Plane

Translations in the coordinate plane involve shifting every point of a shape or graph by the same distance in one or more directions. This process is quite simple and can be visualized as sliding the whole graph horizontally, vertically, or both, without changing its shape or orientation.

• Horizontal Translation: Moves the graph left or right.
• Vertical Translation: Moves the graph up or down.
• Diagonal Translation: Moves the graph in both horizontal and vertical directions.

In our exercise, the graph described by the equation \(x^2 + y^2 = 100\) is a perfect circle centered at the origin (0,0) with a radius of 10. When discussing translations in this context, we refer to adjusting the position of this circle without altering its size or structure. Simply put, the entire circle moves, but remains the same in every other aspect.

Downward Translation

A downward translation involves shifting the graph of a function or equation downward by a certain number of units. This specifically affects the vertical position of the graph on the coordinate plane.

To do a downward translation, you typically adjust the \(y\) variable of the equation. For a translation that moves the graph down by \(k\) units, replace \(y\) with \(y - k\). Here, \(k\) represents the number of units the graph is being shifted down.

In our example, the equation \(x^2 + y^2 = 100\) undergoes a downward translation of 5 units. This transformation requires substituting \(y\) with \(y - 5\), yielding the new equation \(x^2 + (y - 5)^2 = 100\). This adjusts the position of the original circle, moving the center of the circle downward to \( (0, -5) \). The shape and size of the circle itself remain unchanged.

Transformations of Functions

Transformations of functions are modifications that alter the position, size, or shape of a graph. These transformations include translations, reflections, rotations, and dilations. They are fundamental in understanding how algebraic changes affect the graphical representation of a given equation.

• Translations: Shift the graph up, down, left, or right.
• Reflections: Flip the graph over a specified axis.
• Rotations: Turn the graph around a point.
• Dilations: Resize the graph while maintaining shape proportions.

In handling transformations, particularly translations, it is essential to adjust the function's equation accordingly. Translating an equation like \(x^2 + y^2 = 100\) involves revising each instance of \(x\) and/or \(y\) according to the direction and magnitude of the shift. Translating down affects only the \(y\)-value, applying changes as demonstrated in the example discussed. This foundational understanding helps in analyzing and predicting the new positions and appearances of graphs across various transformations.