Let's solve this problem step by step.
In Figure 1 you have the graph of this equation.
In fact, it is an egg curve as you can see, the question is asking for shifting
the curve vertically and horizontally, so what you need to do is the following:
First. Shifting
the curve horizontally (to the left).
Suppose you want to get the curve when the
center of the egg matches with the origin of the Cartesian plane, so you need
to shift the curve to the left. To do this add 40 units to the x-coordinate,
so:
[tex][(x+40)^2+y^2]^2=80(x+40)^3+9(x+40)y^2[/tex]
This shift is shown in Figure 2
Second. Shifting
the curve horizontally (to the right).
Suppose you want to shift the curve 40 units to
the right, then you need to subtract 40 units from the x-coordinate like
this:
[tex][(x-40)^2+y^2]^2=80(x-40)^3+9(x-40)y^2[/tex]
This shift is shown in Figure 3
Third. Shifting the
curve vertically (upward)
Suppose you want to shift the curve 40 units
upward, then you need to subtract 40 units from the y-coordinate like
this:
[tex][x^2+(y-40)^2]^2=80x^3+9x(y-40)^2[/tex]
This shift is shown in Figure 4
Fourth. Shifting the
curve vertically (downward)
Suppose you want to shift the curve 40 units
downward, then you need to add 40 units to the y-coordinate like this:
[tex][x^2+(y+40)^2]^2=80x^3+9x(y+40)^2[/tex]
This shift is shown in Figure 5
