Respuesta :
The equation of the quadratic function is [tex]y=(x-7)(x+3)[/tex]
Explanation:
The vertex form of the quadratic function is given by
[tex]y=a(x-h)^{2}+k[/tex]
It is given that the quadratic function has a vertex at [tex](2,-25)[/tex]
The vertex is represented by the coordinate [tex](h,k)[/tex]
Hence, substituting [tex](h,k)=(2,-25)[/tex] in the vertex form, we get,
[tex]y=a(x-2)^{2}-25[/tex]
Now, substituting the x - intercept [tex](7,0)[/tex] , we have,
[tex]0=a(7-2)^{2}-25[/tex]
[tex]0=a(5)^{2}-25[/tex]
[tex]25=a(25)[/tex]
[tex]1=a[/tex]
Thus, the value of a is 1.
Hence, substituting [tex]a=1[/tex], [tex](h,k)=(2,-25)[/tex] in the vertex form [tex]y=a(x-h)^{2}+k[/tex] , we get,
[tex]y=1(x-2)^{2}-25[/tex]
[tex]y=(x-2)^{2}-25[/tex]
[tex]y=x^2-2x+4-25[/tex]
[tex]y=x^2-2x-21[/tex]
[tex]y=(x-7)(x+3)[/tex]
Thus, the equation of the quadratic function is [tex]y=(x-7)(x+3)[/tex]
