#Daniel Cleary
#April 30, 2018
#Professor Laura Gross
#Sage Application Assignment

show("Exploring the use of Variables in Sage")
show("The Use of Variables in Algebra & Calculus")

show("1.) If a = 3, b = 4, and c = 7, find f(a), f(b), and f(c) if f(x) = x^2 + 2x + 5")

#Variables shown below include a, b, c, and the function f(x).
a = 3
b = 4
c = 7
f(x) = x**2 + 2*x + 5

show("f(a) =")
show(f(a))
show("-----")
#
show("f(b) =")
show(f(b))
show("-----")
#
show("f(c) =")
show(f(c))

show("2.) Graph the same points, a, b, and c, for f(x).")

#The variables will be recycled for the x values, and the y values will be subbed by f(a), f(b), and f(c) to display implementation of variables.
plot(f(x), -12, 10) + point((a,f(a)), size=30) + point((b,f(b)), size=30) + point((c,f(c)), size=30)

show("3.) Let's say d = -x + 3, e = x - 1, f = x + 5. Graph d*e, d*f, and e*f and explain their patterns.")

d(x) = (-x + 3)
e(x) = (x - 1)
f(x) = (x + 5)

show("'d' has a negative leading coefficient, so all graphs containing d are concave down.")
show("All graphs containing 'd' have a zero at 3")
show("All graphs containing 'e' have a zero at 1")
show("All graphs containing 'f' have a zero at -5")
plot(d*e, -15,10) + plot(d*f, -15,10, color='green') + plot(e*f, -15,10, color='red')

show("4.) Plot d*e*f, mark all zero points throughout the graph.")

plot(d*e*f,-10,10) + point((3,d(3)), size=30, color='red') + point((1,e(1)), size=30, color='green') + point((-5,f(-5)), size=30)

show("5.) Find the max and min of the graph displayed in #4.")

show("We know the maxes and mins are the midpoints of each point after finding the zeros, so we can solve by plugging them in.")
show("-----")
show("Min =")
show(d(-2)*e(-2)*f(-2))
show("-----")
show("Max =")
show(d(2)*e(2)*f(2))