Find the range of the quadratic function.
-$$f ( x ) = 2 x ^ { 2 } + 2 x - 5$$
A) $\left[ - \frac { 11 } { 2 } , \infty \right)$
B) $\left( - \infty , - \frac { 11 } { 2 } \right]$
C) $\left[ - \frac { 1 } { 2 } , \infty \right)$
D) $\left( - \infty , - \frac { 1 } { 2 } \right]$

Question

Quizplus · Accepted Answer

We can find the range by completing the square on the quadratic. First, factor out the leading coefficient of 2:

\begin{align*}
f(x) &= 2(x^2+x) -5 \
&= 2(x^2+x+\frac{1}{4}-\frac{1}{4}) -5 \
&= 2 ((x+\frac{1}{2})^2 - \frac{9}{4}) - 5 \
&= 2(x+\frac{1}{2})^2 - \frac{23}{2}
\end{align*}

Since $(x+\frac{1}{2})^2$ is always nonnegative, the range of $f(x)$ is $\left[ - \frac { 23 } { 2 } , \infty ight)$. Solving for the minimum value, when $x=-\frac{1}{2}$, we get $f(-\frac{1}{2})=-\frac{23}{2}+ \frac{23}{2}=0$. Therefore, the range of $f(x)$ is $\left[ - \frac { 11 } { 2 } , \infty ight)$.

Find the Range of the Quadratic Function f(x)=2x2+2x−5f ( x ) = 2 x ^ { 2 } + 2 x - 5f(x)=2x2+2x−5

Find the Range of the Quadratic Function $f ( x ) = 2 x ^ { 2 } + 2 x - 5$