how can the equation (x+2)^2+1 be obtained by applying a horizontial translation to the graph of y=(x-2)^2+1
By obseving that your equation y=(x+2)^2+1 is the same as y=((x+4)-2)^2+1 = y=(x+(4+ (-2))^2+1. Since you're ading to the x value, this is a shift to the left.
To obtain the equation y=(x+2)^2+1 by applying a horizontal translation to the graph of y=(x-2)^2+1, you need to shift the graph 4 units to the left.
Here's how you can do it:
1. Start with the equation y=(x-2)^2+1, which represents the graph shifted 2 units to the right.
2. To shift the graph 4 units to the left, add 4 to the x term inside the parentheses: y=((x+4)-2)^2+1.
3. Simplify the equation: y=(x+2)^2+1.
By adding 4 to the x term inside the parentheses, the graph is horizontally translated to the left. Thus, the equation y=(x+2)^2+1 is obtained from y=(x-2)^2+1 through a horizontal translation.