Given:" "y=-2x^2+2x-3......................(1)
Standard form:" "y=ax^2+bx+c
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color(blue)("Determine y intercept")
color(blue)(y_("intercept")=c=-3
(note: y intercept is at x=0 so -2x^2=0" and "2x=0" leaving " y=-3
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color(blue)("Determine x-vertex")
The coefficient of x^2 is negative so the graph is of general shape nn Thus the vertex is a maximum.
Write " "y=-2x^2+2x-3" as "y=-2(x^2color(red)( -x)) -3
Using the coefficient of color(red)(-x -> -1)
Apply: color(blue)(x_("vertex")=(-1/2)xx(color(red)(-1))=+1/2)
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color(blue)("Determine y-vertex")
Substitute x=1/2 into equation (1) giving
y_("vertex")=-2(1/2)^2+2(1/2)-3
color(blue)(y_("vertex")=-2/4+2/2-3 = -5/2
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color(blue)("Vertex"->(x,y)->(1/2,-5/2))
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color(red)("As the graph is of shape "nn" and the vertex")
color(red)("is below the x-axis there are no x-intercepts")
