Question

The range of the quadratic expression $y=-x^2+3x+3$ is

• A. $(-\infty ,\frac{21}{4}]$
• B. $[-\frac{21}{4},\infty )$
• C. $[\frac{21}{4},\infty )$

Answer 1

The correct option is A $(-\infty ,\frac{21}{4}]$

Given quadratic expression is $y=-x^2+3x+3$
On comparing with standard form of quadratic expression, $y=a{x}^2+bx+c$
we get, $a=-1,b=3,c=3$
$\&D=b^2-4ac=3^2-4\cdot (-1)\cdot 3=21$
$a<0\Rightarrow$ Downward opening parabola.
And it's vertex is given as: $(\frac{-b}{2a},\frac{-D}{4a})$

$\therefore$ Range of the quadratic expression: $(-\infty ,-\frac{D}{4a}]$
$\text{Where}-\frac{D}{4a}=-\frac{21}{4\cdot (-1)}=\frac{21}{4}$
$\therefore \text{Range}\in (-\infty ,\frac{21}{4}]$