What kind of transformation converts the graph of $f(x)=5(x+9)^{2}$ into the graph of $g(x)=$ $5(x+2)^{2}+9 ?$ translation 7 units right and 9 units up translation 7 units left and 9 units up translation 7 units right and 9 units down transiation 7 units left and 9 units down
Solution
Step 1 ：The transformation from \(f(x)=5(x+9)^{2}\) to \(g(x)=5(x+2)^{2}+9\) involves a shift in the x-coordinate and a shift in the y-coordinate.
Step 2 ：The shift in the x-coordinate can be determined by comparing the terms inside the parentheses in \(f(x)\) and \(g(x)\). The original x-coordinate is -9 and the transformed x-coordinate is -2, so the shift in the x-coordinate is 7 units to the right.
Step 3 ：The shift in the y-coordinate can be determined by comparing the constant terms in \(f(x)\) and \(g(x)\). The original y-coordinate is 0 and the transformed y-coordinate is 9, so the shift in the y-coordinate is 9 units up.
Step 4 ：Therefore, the transformation that converts the graph of \(f(x)=5(x+9)^{2}\) into the graph of \(g(x)=5(x+2)^{2}+9\) is a translation 7 units to the right and 9 units up.
Step 5 ：Final Answer: The transformation is a \(\boxed{\text{translation 7 units right and 9 units up}}\).
