Question

In the following exercises, find the minimum or maximum value.

$y=7x^2+14x+6$

Short answer

The minimum value of the function is $-1$and it occurs when$x=-1$.

Step-by-step solution

Given information

The given expression is: $y=7x^2+14x+6$.

Determine the minimum or maximum value.

Comparing the given function with standard form $f\left(x\right)=a{x}^2+bx+c$, the values are $a=7$,$b=14$, $c=6$.

Because the value of the leading coefficient ais positive, so the parabola opens up.

The quadratic function has a minimum.

Determine the equation of axis of symmetry.

The axis of symmetry is the vertical line given by $x=-\frac{b}{2a}$.

Substituting the value of a and b.

$x=-\frac{14}{2\left(7\right)}$

$$\begin{gathered} =-\frac{14}{14} \\ =-1 \end{gathered}$$

The equation of the axis of symmetry is$x=-1$.

Determine the minimum value of function.

Evaluate $y(-1)$by substituting $x=-1$in $y=-x^2+4x-5$.

$$\begin{gathered} y=7{(-1)}^2+14(-1)+6 \\ =7-14+6 \\ =-7+6 \\ =-1 \end{gathered}$$

So, the vertex is $(-1,-1)$.

The minimum value of the function is $-1$and it occurs when $x=-1$.