Question

The graph of quadratic polynomials is ___?

Answer 1

Hint: The graph obtained by plotting a quadratic polynomial comes out to be a Parabola . The roots obtained from solving the quadratic polynomial are the co – ordinates where the parabola cuts the X – Axis . This is because the roots are obtained by putting the given equation equal to zero i.e. where the value of the polynomial is zero .

Complete step-by-step answer:
We know that the standard form of parabola is equals to $$f\left(x\right)=a{x}^2+bx+c$$ , which also a quadratic polynomial where , $$f\left(x\right)$$ is a function of $$x$$ . Now , we have to find the roots of the quadratic equation which can find using the formula, $$\left({\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}\right)$$ , where $$b=$$coefficients of $$x$$ , $$a=$$ coefficients of $$x^2$$ and $$c=$$coefficients of $$c$$ in the given quadratic equation . On putting the values we get the roots as , considering the $$a=b=1,c=-1$$ , we have
$$f\left(x\right)=x^2+x-1$$ , on applying the above formula , we get
$$x={\displaystyle \frac{-1\pm \sqrt{1^2-4\times 1\times (-1)}}{2\times 1}}$$
$$x={\displaystyle \frac{-1\pm \sqrt{1^2+4}}{2}}$$ , on solving we get
$$x={\displaystyle \frac{-1\pm \sqrt{5}}{2}}$$
This is shows it have real and distinct roots which are
$$x_1={\displaystyle \frac{-1+\sqrt{5}}{2}}$$
$$x_1=0.618$$ ( approx. )
  and $$x_2={\displaystyle \frac{-1-\sqrt{5}}{2}}$$
$$x_2=-1.618$$ (approx.)
Now , on plotting the graph of $$f\left(x\right)=x^2+x-1$$ , we get

The intercepts at X – axis are roots of the equation . Also the graph of the equation symbolizes the shape of a parabola .

Note: The graph of a quadratic polynomial depends upon the degree of the polynomial such as if we have two variables with degree as $$2$$ then the shape of the graph will be ellipse , also the roots of a quadratic equation can be of different types which depends on discriminant of Quadratic formula which is $$\sqrt{b^2-4ac}$$ , it is equals to zero we have real and equal roots . If it is greater than zero we have real and unequal roots . If the discriminant is less than zero we have complex roots .