What causes parabolas to shift side to side or up and down?

1 Answer
Dec 21, 2017

See explanation

Explanation:

#color(blue)("Summery")#

Given #y=ax^2+bx+c# the graph must always pass through through the point #y_("intercept")->(x,y)=(0,c)# To satisfy this if the graphs of form #uu# is 'moved' sideways it lowers. If the graph of form #nn# then it rises. What make the graph move' sideways is the #bx# part of #y=ax^2+bx+c #
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#color(white)("d")#

Consider again the standardised form #y=ax^2+bx+c#

Note that for graph of type "#nn or uu# the maximum or minimum is called the vertex.

You can also have a quadratic of the forms #sub or sup# These still have a vertex but it is the left most point for #sub# and the right most point for #sup#. You will not come across these for quite a while.
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#color(white)("d")#

# color(blue)("The up or down is linked to the left or right")#

#color(brown)("The main point")#
It all depends an the values of #b and c " in " y=ax^2+bx+c#
The graph always passes through the point where #x=0 and y=c#

#P_("y-intercept")->(x,y)=(0,c) #

As the graph must always pass through this point on the y-axis it influences the vertical 'movement' of the plot if we move the plot sideways. Suppose #c > 0# then in the case of shape #uu# the graph will lower. In the case of the shape #nn# the graph will rise.
If #c<0 # then what happens is the other way round.

#color(brown)("The amount of sideways movement for type "nn or uu)#

Given that we have #y=ax^2+bc+c# write it as #y=a(x^2+b/ax)+c#

The #x# value for lowest point of type #uu# and the highest point of type #nn# (vertex) is determined by:

#x_("vertex")=(-1/2)xxb/a#

Consider the example equation of #y=2x^2+8x+5color(white)("d") color(green)("(green plot)")#
Where #a=2; b=8 and c=5#

#y_("intercpt")->(x,y)=(0,5)#
#x_("vertex")=(-1/2)xxb/acolor(white)("d")=color(white)("d")(-1/2)xx8/2color(white)("d")=color(white)("d")-2#

Tony B