Question

What is an equation for the translation of by 3 units left and 5 units up? (x − 2)^2 + (y + 1)^2 =16

Answer 1

The translated equation of the circle originally given by (x − 2)^2 + (y + 1)^2 = 16, when moved 3 units left and 5 units up, is (x + 1)^2 + (y - 4)^2 = 16. When we subtract k from y then the circle is translated up k units. When we add k to y then the circle is translated down k units. As the radius increases the size of the circle also increases. A circle can be horizontally translated by increasing or decreasing the x-values by a constant number.

Step-by-step explanation:

The equation (x − 2)^2 + (y + 1)^2 = 16 represents a circle centered at (2, -1) with a radius of 4. To translate this circle by 3 units left and 5 units up, we change the center of the circle accordingly.

To move the circle 3 units to the left, we add 3 to the x-coordinate of the center, changing it from (2, -1) to (-1, -1). To move it 5 units up, we subtract 5 from the y-coordinate, changing it from (-1, -1) to (-1, 4). Hence, the new equation of the translated circle is (x + 1)^2 + (y - 4)^2 = 16.

1. Identify the center of the original circle: (2, -1).
2. Determine the direction of translation: 3 units left (which is -3 units on the x-axis) and 5 units up (which is +5 units on the y-axis).
3. Translate the center of the circle: Original center (2, -1) - New center (-1, 4).
4. Write the new equation using the translated center: (x + 1)^2 + (y - 4)^2 = 16.

This new equation represents the original circle translated to its new position.