Question

Find the minimum y-value on the graph of y=f(x)

F(x) = 7x^2 + 7 - 6

Answer 1

The minimum y-value is

`y=-(31)/(4)` or
`y=-7.75`

Explanation:

we have

`f(x)=7x^(2)+7x-6`

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

`f(x)+6=7x^(2)+7x`

Factor the leading coefficient

`f(x)+6=7(x^(2)+x)`

Complete the square. Remember to balance the equation by adding the same constants to each side

`f(x)+6+1.75=7(x^(2)+x+0.25)`

`f(x)+7.75=7(x^(2)+x+0.25)`

Rewrite as perfect squares

`f(x)+7.75=7(x+0.5)^(2)`

`f(x)=7(x+0.5)^(2)-7.75` ------> 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
`y=-7.75=-7(3)/(4)=-(31)/(4)`

see the attached figure to better understand the problem