Question

Select the correct answer. Which graph is represented by this equation? 5x+5y^2+40y=0

Answer 1

The equation
`5x + 5y^2 + 40y = 0` represents a parabola that opens to the left with a vertex at (-4, 0) when rearranged in standard form.

The equation given is
`5x + 5y^2 + 40y = 0`. To understand which graph is represented by this equation, we can rearrange it to complete the square for y. Here are the steps:

First, factor out the 5 from the y terms:
`5(x + y^2 + 8y) = 0.`

Now, complete the square on the y terms:
`y^2 + 8y + 16` is a perfect square trinomial
`((y+4)^2).`

Subtract 16 from both sides to keep the equation balanced:
`5(x + (y+4)^2 - 16) = 0.`

Rewrite the equation:
`5x + 5(y+4)^2 = 80.`

Divide by
`5: x + (y+4)^2 = 16.`

Finally, rearrange to reveal the form of a shifted circle:
`(y+4)^2 = 16 - x.`

This is the equation of a parabola that opens to the left with vertex at (-4, 0) and a horizontal axis of symmetry. So, the correct graph would be that of a leftward-opening parabola.