Given:
#y=-x^2+3x-3# is a quadratic equation in standard form:
#ax^2+bx+c#,
where:
#a=-1#, #b=3#, and #c=-3#
Axis of symmetry: vertical line that divides the parabola into two equal halves, and is also the #x#-value of the vertex. For a quadratic equation, #x=(-b)/(2a)#.
#x=(-3)/(2*(-1))#
#x=(-3)/-2#
#x=3/2# #larr# Axis of symmetry
Vertex: the maximum or minimum point #(x,y)# of a parabola. Since #a<0#, the vertex will be the maximum point and the parabola opens downward.
Substitute #3/2# for #x# in the equation and solve for #y#.
#y=-(3/2)^2+3(3/2)-3#
Simplilfy.
#y=-9/4+9/2-3#
Multiply each number by a fraction equal to #1# so that each of their denominators is #4#. For example, #6/6=1#. Recall that a whole number is understood to have a denominator of #1#.
#y=-9/4+9/2xxcolor(teal)(2/2)-3/1xxcolor(magenta)(4/4#
Simplify.
#y=-9/4+18/4-12/4#
#y=(-9+18-12)/4#
#y=-3/4#`
Vertex: #(3/2,-3/4)# or #(1.5,-0.75)#
X-intercepts: The parabola does not cross the x-axis, so there are no real solutions, so there are no intercepts, but there are complex solutions.
Substitute #0# for #y#. Use the quadratic formula to solve for #x#.
#0=-x^2+3x-3#
Quadratic Formula
#x=(-b+-sqrt(b^2-4ac))/(2a)#
Plug in the known values.
#x=(-3+-sqrt(3^2-4*-1*-3))/(2*-1)#
Simplify.
#x=(-3+-sqrt(-3))/(-2)#
Complex solutions for #x#.
#x=(3+sqrt(3)i)/(2),##(3-sqrt(3)i)/(2)#
graph{y=-x^2+3x-3 [-16.08, 15.94, -13.46, 2.56]}
