Question

Which among the following is the correct graphical representation of the quadratic polynomial $y=-2x^2+2x-0.5?$

Answer 1

Given: $y=-2x^2+2x-0.5$
Since the coefficient of $x^2$ is negative
$\Rightarrow$ The parabola will be downward opening parabola.
On comparing with standard form of quadratic expression $\text{y}=a{x}^2+bx+c,$ we get:
$a=-2,b=2,c=-\frac{1}{2}$

$\&D=b^2-4ac=(2{)}^2-4\cdot (-2)\cdot (-\frac{1}{2})=0$

Here, $D=0$ therefore the given quadratic expression has repeated real root.
Also, we get the roots of the quadratic equation by putting $y=0$
$\Rightarrow y=-2x^2+2x-\frac{1}{2}=0$

$\Rightarrow -2x^2+x+x-\frac{1}{2}=0$

$\Rightarrow -2x(x-\frac{1}{2})+(x-\frac{1}{2})=0$

$\Rightarrow (x-\frac{1}{2})(1-2x)=0$

$\Rightarrow x=\frac{1}{2},\frac{1}{2}$
From this we can say that parabola touches $x$-axis at $(\frac{1}{2},0).$