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

determine the vertex form and the maximum or minimum value of the function.\n\n$f(x)=2x^{2}+8x + 3$\n\na $f(x)=2(x + 4)^{2}-13$, the maximum value of $f$ is $-13$\n\nb $f(x)=2(x + 4)^{2}-13$, the minimum value of $f$ is $-13$\n\nc $f(x)=2(x + 2)^{2}-5$, the maximum value of $f$ is $-5$\n\nd $f(x)=2(x + 2)^{2}-5$, the minimum value of $f$ is $-5$

determine the vertex form and the maximum or minimum value of the function.\n\n$f(x)=2x^{2}+8x + 3$\n\na $f(x)=2(x + 4)^{2}-13$, the maximum value of $f$ is $-13$\n\nb $f(x)=2(x + 4)^{2}-13$, the minimum value of $f$ is $-13$\n\nc $f(x)=2(x + 2)^{2}-5$, the maximum value of $f$ is $-5$\n\nd $f(x)=2(x + 2)^{2}-5$, the minimum value of $f$ is $-5$

Answer

Answer:

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

Explanation:

Step1: Complete the square

For the 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 $(x + m)^{2}=x^{2}+2mx+m^{2}$, here $2m = 4$ (so $m = 2$) and $x^{2}+4x=(x + 2)^{2}-4$. Then $f(x)=2((x + 2)^{2}-4)+3$.

Step2: Simplify the expression

Expand the right - hand side: $f(x)=2(x + 2)^{2}-8 + 3=2(x + 2)^{2}-5$.

Step3: Determine the maximum or minimum

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