Respuesta :
We know that the vertex form of any quadratic function is the form:
[tex] f (x) = a(x - h)^2 + k [/tex]
The given quadratic function is [tex]f(x) = 4x^2 + 16x - 9 [/tex]
To convert it into the vertex form we will have to complete the square of the quadratic expression given here which may be done as shown below:
[tex] f(x) = 4x^2 + 16x - 9 =4(x^2+4x)-9 [/tex]
Thus, [tex] f(x)=4(x^2+4x+\mathbf{4})-\boldsymbol{\mathbf{16}}-9 [/tex] (This is because adding a 4 inside the parentheses to complete the square is equivalent to adding 16 to the whole expression and thus we need to subtract 16 from the overall expression.)
[tex] \therefore f(x)=4(x+2)^2-25 [/tex]
The above is the vertex form of the original given function, [tex]f(x) = 4x^2 + 16x - 9 [/tex].
