Answer:
The minimum y-value is
[tex]y=-\frac{31}{4}[/tex] or [tex]y=-7.75[/tex]
Step-by-step explanation:
we have
[tex]f(x)=7x^{2}+7x-6[/tex]
This is the equation of a vertical parabola open up
The vertex is the minimum y-value on the graph
Convert the equation into vertex form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]f(x)+6=7x^{2}+7x[/tex]
Factor the leading coefficient
[tex]f(x)+6=7(x^{2}+x)[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]f(x)+6+1.75=7(x^{2}+x+0.25)[/tex]
[tex]f(x)+7.75=7(x^{2}+x+0.25)[/tex]
Rewrite as perfect squares
[tex]f(x)+7.75=7(x+0.5)^{2}[/tex]
[tex]f(x)=7(x+0.5)^{2}-7.75[/tex] ------> equation in vertex form
The vertex is the point (-0.5,-7.75)
therefore
The minimum is the point (-0.5,-7.75)
The minimum y-value is [tex]y=-7.75=-7\frac{3}{4}=-\frac{31}{4}[/tex]
see the attached figure to better understand the problem
