Question

translations rule!
try each rule. click the button for each choice.
select any one of them and explain how the algebra of the rule connects to the translation of the triangle.
\\( (x,y) \to (x + 6,y + 4) \\)
\\( (x,y) \to (x + 6,y - 4) \\)
\\( (x,y) \to (x - 6,y + 4) \\)
\\( (x,y) \to (x - 6,y - 4) \\)

Explanation:

Step1: Choose a translation rule

Let's choose the rule \((x,y) \to (x + 6,y + 4)\).

Step2: Understand the translation components

In a coordinate plane, the \(x\)-coordinate represents the horizontal position (left - right) and the \(y\)-coordinate represents the vertical position (up - down). When we have a translation rule \((x,y)\to(x + a,y + b)\), the value of \(a\) tells us the horizontal translation (positive \(a\) means moving to the right, negative \(a\) means moving to the left) and the value of \(b\) tells us the vertical translation (positive \(b\) means moving up, negative \(b\) means moving down).

For the rule \((x,y)\to(x + 6,y + 4)\), the \(x\)-coordinate of each vertex of the triangle is increased by 6. This means that each point on the triangle will move 6 units to the right horizontally. The \(y\)-coordinate of each vertex is increased by 4, which means that each point on the triangle will move 4 units up vertically.

For example, if we take a vertex of the triangle with coordinates \((x_1,y_1)\), after applying the rule, its new coordinates will be \((x_1+6,y_1 + 4)\). So, the entire triangle will be translated 6 units to the right and 4 units up on the coordinate plane.

Answer:

We chose the rule \((x,y) \to (x + 6,y + 4)\). The algebra of the rule connects to the translation as follows: The \(x+6\) part of the rule indicates that each point of the triangle moves 6 units to the right (since adding a positive number to the \(x\)-coordinate in a coordinate system moves the point to the right), and the \(y + 4\) part indicates that each point moves 4 units up (since adding a positive number to the \(y\)-coordinate moves the point up). This algebraic manipulation of the \(x\) and \(y\) coordinates directly corresponds to the geometric translation of the triangle 6 units right and 4 units up.