Question

The graph of \(f(x)=-x^2+x-2\) is

Answer 1

The given polynomial \(f(x)=-x^2+x-2\) compairing it with \(f(x)=ax^2+bx+c\)
We have
\(a=-1,~b=1,~c=-2\)
\(\therefore D=1^2-4\cdot(-2)\cdot (-1)=-7\\\Rightarrow D<0\)
Hence, roots are not real.
Thus, graph will not intersect with \(x-\) axis and the graph of \(y=f(x)\) will be downward parabola as $a<0$ and intersect the \(y-\)axis at \((0,-2)\) because \(c=-2\)

Now, the coordinates of vertex is \(\left(-\dfrac b{2a},-\dfrac D{4a}\right)\)
\(\Rightarrow -\dfrac b{2a}=\dfrac12,~-\dfrac D{4a}=-\dfrac74\)
\(\therefore\left(-\dfrac b{2a},-\dfrac D{4a}\right)\equiv\left(\dfrac12,-\dfrac74\right)\)
Therefore, graph will be