Answer :
Answer: The answer is f(x) = - 2x²+12x-22.
Step-by-step explanation: We are given to write the function describing a parabola with vertex (3, -4) and passing through the point (5, -12).
We know that the standard form of a parabola with vertex (h, k) is given by
[tex]f(x)-k=a(x-h)^2.[/tex]
Here (h, k) = (3, -4), so we have
[tex]f(x)-(-4)=a(x-3)^2\\\\\Rightarrow f(x)+4=a(x-3)^2.~~~~~~~~~~~(I)[/tex]
Also, the parabola is passing through the point (5, -12), so
[tex]-12+4=a(5-3)^2\\\\\Rightarrow -8=a\times 4\\\\\Rightarrow a=-2.[/tex]
Substituting the value of 'a' above in equation (I), we have
[tex]f(x)+4=-2(x-3)^2\\\\\Rightarrow f(x)=-2(x^2-6x+9)-4\\\\\Rightarrow f(x)=-2x^2+12x-22.[/tex]
Thus, the answer is f(x) = - 2x²+12x-22..