The structure of this is such that it implies f(y) instead of f(x) so that is how I am going to treat it. The consequence is that it rotates an f(x) graph so that the axis of symmetry is parallel to the x-axis instead of the y-axis.
Write as:
" "4x=y^2-6y+1
Divide both sides by 4
" "x=1/4 y^2-6/4y+1/4
color(blue)("Determine the vertex")
Write as:
" "x=1/4(y^2-6y)+1/4
y_("vertex")=(-1/2)xx(-6) = +3
x_("vertex")=1/4(3)^2-3/2(3)+1/4" "=" "-2
color(blue)("Vertex "->(x,y)->(-2,3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determine "x" intercept")
Set y=0
color(blue)(=> x_("intercept")=1/4(0)^2-3/2(0)+1/4)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determine "y" intercept")
Set x=0
=>1/4(y^2-6y)+1/4=0
color(brown)("Completing the square")
1/4(y-3)^2+1/4 +k=0
Where k is the constant of correction.
k=(-1)xx[1/4(-3)^2] = -9/4 giving:
color(brown)(1/4(y-3)^2+1/4 +k=0)color(blue)(" "->" "1/4(y-3)^2-2=0
'....................................................................
=>(y-3)^2=8
y-3=+-sqrt(8)" "=+-2sqrt(2)
y_("intercept")=3+-2sqrt(2)
color(blue)(y_("intercepts")~~5.828" and "0.172" to 3 decimal places")
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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