$5.$ Suppose a quadratic function is given by y $=$ f$($x$)=$ x$29$x $+8.$

$($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. $[5]$

$($b$)$ Find the roots of the quadratic function. $[5]1$

$($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$29$x $+21.$ How many roots does this parabola have? $[10]$

$6.$ Suppose an inverse demand function is given by p $=10$ qd and an inverse supply function is given by p $=4+2$qs$.$

$($a$)$ What is the equilibrium price and quantity combination? $[5]$

$($b$)$ What happens when the price is above or below the equilibrium price? $[5]$

$7.$ How many solutions does a system of two equations have? Explain clearly and succinctly. $[10]$