Question 971114
<pre>
y = -x^2+3x-4

compare to:

y = ax^2+bx+c

a=-1, b=3, c=-4

1. It has an x^2 term, so the graph of the equation is a parabola.
2. The coefficient of the x^2 term, a=-1, is negative so the parabola opens downward.
3. The constant term c is -4, so the y-intercept is (0,c) = (0,-4).
4. To find the vertex we use the vertex formula for the x-coordinate of the vertex,
which is {{{h}}}{{{""=""}}}{{{-b/(2a)}}}{{{""=""}}}{{{(-3)/((2)(-1))}}}{{{""=""}}}{{{3/2}}}, that also tells us that the line of symmetry 
has equation {{{x=3/2}}}
5. The y-coordinate of the vertex is found by substituting the x-coordinate
of the vertex, h, for x in the original equation:
{{{y}}}{{{""=""}}}{{{-(3/2)^2+3(3/2)-4}}}{{{""=""}}}{{{-9/4+9/2-4}}}{{{""=""}}}{{{-9/4+18/4-16/4}}}{{{-7/4)}}}
6. The vertex is below the x-axis and the parabola opens downward so there
are no x-intercepts, and there is no use to try to find any.
7. The vertex is the point (h,k) = {{{(matrix(1,3,3/2,",",-7/4))}}}{{{""=""}}}{{{(matrix(1,3,1&1/2,",",-1&3/4))}}}
8. We plot the vertex, the y-axis and the line of symmetry (in green):

{{{drawing(2800/17,400,-2,5,-14,3,

graph(2800/17,400,-2,5,-14,3), green(line(3/2,-20,3/2,20)),

circle(1.5,-1.75,0.15),circle(1.5,-1.75,0.13),circle(1.5,-1.75,0.11),circle(1.5,-1.75,0.09),circle(1.5,-1.75,0.07),circle(1.5,-1.75,0.05),circle(1.5,-1.75,0.03),circle(1.5,-1.75,0.01),

circle(0,-4,0.15),circle(0,-4,0.13),circle(0,-4,0.11),circle(0,-4,0.09),circle(0,-4,0.07),circle(0,-4,0.05),circle(0,-4,0.03),circle(0,-4,0.01) )}}}

9. Using the line of symmetry, we find the point symmetrical to the y-intercept,
which is (3,-4) on the other side of the line of symmetry:

{{{drawing(2800/17,400,-2,5,-14,3,

graph(2800/17,400,-2,5,-14,3), green(line(3/2,-20,3/2,20)),

circle(1.5,-1.75,0.15),circle(1.5,-1.75,0.13),circle(1.5,-1.75,0.11),circle(1.5,-1.75,0.09),circle(1.5,-1.75,0.07),circle(1.5,-1.75,0.05),circle(1.5,-1.75,0.03),circle(1.5,-1.75,0.01),

circle(3,-4,0.15),circle(3,-4,0.13),circle(3,-4,0.11),circle(3,-4,0.09),circle(3,-4,0.07),circle(3,-4,0.05),circle(3,-4,0.03),circle(3,-4,0.01),



circle(0,-4,0.15),circle(0,-4,0.13),circle(0,-4,0.11),circle(0,-4,0.09),circle(0,-4,0.07),circle(0,-4,0.05),circle(0,-4,0.03),circle(0,-4,0.01) )}}}

10. And now it's easy to sketch in the parabola:

{{{drawing(2800/17,400,-2,5,-14,3,

graph(2800/17,400,-2,5,-14,3,-x^2+3x-4), green(line(3/2,-20,3/2,20)),

circle(1.5,-1.75,0.15),circle(1.5,-1.75,0.13),circle(1.5,-1.75,0.11),circle(1.5,-1.75,0.09),circle(1.5,-1.75,0.07),circle(1.5,-1.75,0.05),circle(1.5,-1.75,0.03),circle(1.5,-1.75,0.01),

circle(3,-4,0.15),circle(3,-4,0.13),circle(3,-4,0.11),circle(3,-4,0.09),circle(3,-4,0.07),circle(3,-4,0.05),circle(3,-4,0.03),circle(3,-4,0.01),



circle(0,-4,0.15),circle(0,-4,0.13),circle(0,-4,0.11),circle(0,-4,0.09),circle(0,-4,0.07),circle(0,-4,0.05),circle(0,-4,0.03),circle(0,-4,0.01) )}}}

Edwin</pre>