Certainly! Let's address the transformation of the given function [tex]\( y = x^2 \)[/tex]:
1. Reflected across the x-axis:
To reflect [tex]\( y = x^2 \)[/tex] across the x-axis, we multiply the function by [tex]\(-1\)[/tex]. Therefore, the reflection of [tex]\( y = x^2 \)[/tex] across the x-axis is:
[tex]\[
y = -x^2
\][/tex]
2. Shifted right 3 units:
To shift any function [tex]\( y = f(x) \)[/tex] to the right by 3 units, we replace [tex]\( x \)[/tex] with [tex]\( x - 3 \)[/tex] in the function. Let’s apply this to our reflected function [tex]\( y = -x^2 \)[/tex]:
[tex]\[
y = -(x - 3)^2
\][/tex]
3. Shifted up 6 units:
To shift any function [tex]\( y = f(x) \)[/tex] up by 6 units, we add 6 to the function. Applying this to the function [tex]\( y = -(x - 3)^2 \)[/tex]:
[tex]\[
y = -(x - 3)^2 + 6
\][/tex]
Therefore, the equation for the function that has the graph with the shape of [tex]\( y = x^2 \)[/tex], but reflected across the x-axis, shifted right by 3 units, and shifted up by 6 units is:
[tex]\[
y = -(x - 3)^2 + 6
\][/tex]
