determine the vertex form and the maximum or minimum value of the function.\n\n$f(x)=2x^{2}+8x + 3$\n\na…

Explanation:

Step1: Convert to vertex - form

For a quadratic function $y = ax^{2}+bx + c$, the vertex - form is $y=a(x - h)^{2}+k$. We complete the square for $f(x)=2x^{2}+8x + 3$. First, factor out the coefficient of $x^{2}$ from the first two terms: $f(x)=2(x^{2}+4x)+3$. Then, complete the square inside the parentheses. For the 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 $f(x)=2((x + 2)^{2}-4)+3$. Expand the expression: $f(x)=2(x + 2)^{2}-8 + 3=2(x + 2)^{2}-5$.

Step2: Determine maximum or minimum

Since $a = 2>0$ for the quadratic function $y = 2(x + 2)^{2}-5$, the parabola opens upward. The vertex of the parabola in vertex - form $y=a(x - h)^{2}+k$ is $(h,k)$. Here, $h=-2$ and $k = - 5$. The vertex is $(-2,-5)$, and the function has a minimum value at the vertex. The minimum value of the function $f(x)$ is $-5$.

Answer:

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