Question

Find the minimum y-value on the graph of y=f(x) F(x) = 7x^2 + 7 - 6

Answer 1

Answer:

The minimum y-value is

$y=-\frac{31}{4}$  or  $y=-7.75$

Step-by-step 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 $y=-7.75=-7\frac{3}{4}=-\frac{31}{4}$

see the attached figure to better understand the problem