Exercise

The graph of $y=-5x^2+2x-4$

Consider a quadratic polynomial $f\left(x\right)=a{x}^2+x+c$ such that $ac>\frac{1}{8}$ and its graph completely lies below $x$-axis, then

Sketch the graph of the function $f\left(x\right)=-0.01x^2+0.5x+2.56$

Draw the graph of the polynomial $p\left(x\right)=x^2-2x-8$.

Draw the graph of the given polynomial and find the zeroes. Justify the answer.

$p\left(x\right)=x^2-1$

If the graph of the polynomial $f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+ . . . +a_n=0$ passes through the origin, then prove that $a_n=0$.

In the figure, the graph of $f\left(x\right)=a{x}^2+bx+1$ is shown.  Write the signs of $a$ and $b^2-4ac$.

In the figure the graph of $f\left(x\right)=a{x}^2+bx+c$ is shown. What is the value of $b^2-4ac$ and the number of real zeroes of $f\left(x\right)$.