To determine the equation for translating [tex]\(x^2 + y^2 = 64\)[/tex] three units to the left and two units down, we need to adjust the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] accordingly.
1. Translate three units to the left:
- Translating a function three units to the left involves replacing [tex]\(x\)[/tex] with [tex]\(x + 3\)[/tex].
- Therefore, in the original equation [tex]\(x^2 + y^2 = 64\)[/tex], replace [tex]\(x\)[/tex] with [tex]\(x + 3\)[/tex]:
[tex]\[
(x + 3)^2 + y^2 = 64
\][/tex]
2. Translate two units down:
- Translating a function two units down involves replacing [tex]\(y\)[/tex] with [tex]\(y + 2\)[/tex] (since moving down the y-axis decreases the value of [tex]\(y\)[/tex]).
- Therefore, in the equation [tex]\((x + 3)^2 + y^2 = 64\)[/tex], replace [tex]\(y\)[/tex] with [tex]\(y + 2\)[/tex]:
[tex]\[
(x + 3)^2 + (y + 2)^2 = 64
\][/tex]
Thus, the correct equation for the translation of [tex]\(x^2 + y^2 = 64\)[/tex] three units to the left and two units down is:
[tex]\[
(x + 3)^2 + (y + 2)^2 = 64
\][/tex]
So, the correct answer is:
[tex]\[
(x + 3)^2 + (y + 2)^2 = 64
\][/tex]
