Answer:
[tex]f^\prime(x)=-(x+1)^2-4[/tex]
Step-by-step explanation:
We have the function:
[tex]f(x)=x^2[/tex]
And we want to write its equation after being: 1) Flipped over the x-axis, 2) shifted 4 units down, and 3) shifted 1 unit to the left.
To denote a flip over the x-axis, we multiply the function by -1. Hence:
[tex]f^\prime(x)=-x^2[/tex]
Is our function flipped over the x-axis.
To shift n units vertically, we simply add n to our function.
Since we are going 4 units downwards, we will add -4 to our function. Hence:
[tex]f^\prime(x)=-x^2-4[/tex]
Finally, to shift n units horizontally, we substitute our variable [tex]x[/tex] for [tex](x-n)[/tex].
Since we are shifting 1 unit to the left, n=-1. Hence, we will substitute [tex]x[/tex] for [tex](x+1)[/tex]. Therefore:
[tex]f^\prime(x)=-(x+1)^2-4[/tex]
So, our final function is:
[tex]f^\prime(x)=-(x+1)^2-4[/tex]
