determine the vertex form and the maximum or minimum value of the function. f(x)=2x² + 8x + 3 a f(x)=2(x +…

Explanation:

Step1: Complete the square

For a quadratic function $y = ax^{2}+bx + c$, the vertex - form is $y=a(x - h)^{2}+k$. Given $f(x)=2x^{2}+8x + 3$, factor out the coefficient of $x^{2}$ from the first two terms: $f(x)=2(x^{2}+4x)+3$. Complete the square inside the parentheses. For the quadratic expression $x^{2}+4x$, we know that $(x + m)^{2}=x^{2}+2mx+m^{2}$, and if $2m = 4$, then $m = 2$ and $x^{2}+4x=(x + 2)^{2}-4$. So $f(x)=2((x + 2)^{2}-4)+3$.

Step2: Simplify the expression

Expand the right - hand side: $f(x)=2(x + 2)^{2}-8 + 3=2(x + 2)^{2}-5$. Since $a = 2>0$, the parabola opens upward. For a parabola $y=a(x - h)^{2}+k$ with $a>0$, the vertex is $(h,k)$ and the function has a minimum value at the vertex. Here $h=-2,k = - 5$, so the minimum value of $f(x)$ is $-5$.

Answer:

D. $f(x)=2(x + 2)^{2}-5$, the minimum value of $f$ is $-5$