Suppose a quadratic function is given by y $=$ f$($x$)=$ x$\mathrm{²}$ − 9x $+8.$

$($a$)$ Does this function represent an upward or downward facing parabola? Explain. Find the x$-$coordinate of the vertex $($or the equation of the vertical line that intersects the parabola at the vertex$)$ of the parabola. 

$($b$)$ Find the roots of the quadratic function. 

$($c$)$ Using information from $($a$)$ and $($b$)$ draw a well labelled graph of the parabola $($in particular, label the vertex and the roots$).[5]$

$($d$)$ In the same graph as $($c$),$ draw the parabola if it was instead represented by the function f$($x$)=$ x$2-9$x $+21.$ How many roots does this parabola have? $[10]$