1. Consider the graph of f (x) =x2 -4x+5 a. Graph f(x) in detail in the xy plane of the coordinate system provided on the next page without using a graphing calculator. b. What conclusion can you make about the roots of f (x)? S No real roots 1 ( 2, 15 - c. Show that 2 - i is a root for f (x). Put in F ( 2 - 1)=0 equation F ( 2 - 1 ) = ( 2 - 1 ) - 4 ( 2 - ; ) +5 d. Show that f (2 -37) is a real number. ( 2 -1 ) ( 2- 1)8 +4its (2-37 ) (2- 3:) a(8 4- 2iti2- 8 +5+ 4: - 8 + 12:+5 4-6i-6it9i-3+12:- 21 e. Show that f (3+i) is not a real number. F ( 3 + 1 ) = (3+ 1) -4 (341)+5- 9+31+3i+12-7-41 4- 1- 8 +5 (3+:)(3+0)-12-4:+5 9- 9 = 0 f. Try a few more complex values and make a conjecture about values of a and b for which f(a + bi) is a real number. Explain how you arrived at your conjecture and prove that it is true. g. Lastly, draw a graphical representation of what your above answer imply about the real-valued outputs of fwith regard to the inclusion of a complex domain. y Complex values have solutions on the i-axis imaginary axis X