Question

I am doing a project in my math class where we create a picture with functions and needed an egg curve. I finally figured out the perfect equation but don't know how to shift the curve vertically and horizontally. The equation I have now is (x^2+y^2)^2=(80)x^3+(9)xy^2

Answer 1

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:

$[(x+40)^2+y^2]^2=80(x+40)^3+9(x+40)y^2$

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:

$[(x-40)^2+y^2]^2=80(x-40)^3+9(x-40)y^2$

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:

$[x^2+(y-40)^2]^2=80x^3+9x(y-40)^2$

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:

$[x^2+(y+40)^2]^2=80x^3+9x(y+40)^2$

This shift is shown in Figure 5