$($a$)$ What is the vertex of the parabola given by $y=\frac{1}{2}(x-3{)}^2?$

$($b$)$ Which of the following equations represents a power function? $[$Select all that apply.$]$

i$.y=\frac{5x^3}{x}$

iii. $g(x)=-x^3\sqrt[\phantom{2}]{x}$

ii$.f(x)=(2-x{)}^2$

iv$.y-3x^3=0$

$($c$)$ What is the $y-$intercept of $f(x)=(3-x)(1+2x)(x-1)?$

$($d$)$ True or false? The graph of $f(x)=(4-x)(2-3x)$ is an upward opening parabola.

$($e$)$ Which of the following polynomials have $x=2$ as a root? $[$Select all that apply.$]$

i$.y=x^2-5$

iii. $f(x)=x^4-3x^3+x+6$

ii$.f(x)=(3x-1)(2-x)(x+1)$

iv$.y=(x-3{)}^3?$

$($f$)$ What is the maximum number of turning points in the graph of $5)(x-3{)}^2(2x^2+1)?$