Answer: [tex]f(x)=-4(x+\frac{3}{4})^2+\frac{13}{4}[/tex]
Step-by-step explanation:
The Vertex form of a quadratic function is [tex]f(x)=a(x-h)^2+k[/tex], where [tex](h,k)[/tex] is the vertex of the parabola, and the sign of the coefficient [tex]a[/tex] indicates if the parabola opens down or opens up.
The Standard form of a quadratic function is [tex]f(x)=ax^2+bx+c[/tex], where the sign of the coefficient [tex]a[/tex] indicates if the parabola opens down or opens up.
To transform a quadratic function from Standard form to Vertex form, you need to complete the square.
Given the quadratic function [tex]f(x)=-4x^{2}-6x+1[/tex]:
Indentify a:
[tex]a=-4[/tex]
Factor out -4:
[tex]f(x)=-4(x^2+\frac{3}{2}+\frac{1}{4})[/tex]
Take the coefficient b and add and subtract [tex](\frac{b}{2})^2[/tex] to keep the balance:
[tex](\frac{b}{2})^2=(\frac{\frac{3}{2}}{2})^2=(\frac{3}{4})^2[/tex]
[tex]f(x)=-4(x^2+\frac{3}{2}x+(\frac{3}{4})^2)-\frac{1}{4}-(\frac{3}{4})^2)[/tex]
[tex]f(x)=-4(x^2+\frac{3}{2}x+(\frac{3}{4})^2)) +1+(\frac{9}{4})[/tex]
Now rewrite the expression in the form [tex]-4(x+\frac{b}{2})^2[/tex], this is:
[tex]f(x)=-4(x+\frac{3}{4})^2+\frac{13}{4}[/tex]
