Respuesta :
The expression we have is:
[tex]f\mleft(x\mright)=-2x^2+28x-99[/tex]Since this is a quadratic equation, is the equation of a parabola in standard form:
[tex]f(x)=ax^2+bx+c[/tex]In this case, comparing the general form and the equation we have:
[tex]\begin{gathered} a=-2 \\ b=28 \\ c=-99 \end{gathered}[/tex]We will define the vertex of the parabola as (h,k). And the general vertex form is:
[tex]f(x)=a(x-h)^2+k[/tex]Where h is defined as follows:
[tex]h=\frac{-b}{2a}[/tex]We already know b and a, so we substitute them to find h:
[tex]h=\frac{-28}{2(-2)}=\frac{-28}{-4}=7[/tex][tex]h=7[/tex]Now we only need to find k, which is defined as the value of the function when x=h:
[tex]k=f(h)[/tex]So we find f(h) by substituting h=7 into the original expression:
[tex]\begin{gathered} f(x)=-2x^2+28x-99 \\ k=f(h)=-2(7)^2+28(7)-99 \end{gathered}[/tex]Solving the operations to find k:
[tex]\begin{gathered} k=-2(49)+196-99 \\ k=-98+196-99 \\ k=-1 \end{gathered}[/tex]Now that we have h and k, we go back to the general vertex form:
[tex]f(x)=a(x-h)^2+k[/tex]And substitute a=-2, h=7 and k=-1:
[tex]f(x)=-2(x-7)^2-1[/tex]Answer:
[tex]f(x)=-2(x-7)^2-1[/tex]