Final answer:
To sketch the graph of the equation 2x + y² + 8y + 8 = 0, we complete the square for the y-terms and find that the equation represents a sideway parabola opening to the left with its vertex at (-4, -4).
Step-by-step explanation:
To sketch the graph of the equation 2x + y² + 8y + 8 = 0, we will first complete the square for the y-terms to help in identifying the type of graph we are dealing with. We look to rewrite the equation in a form that resembles the equation of a conic section.
Rewrite the y-terms:
y² + 8y = (y + 4)² - 16
Substitute back into the original equation:
2x + (y + 4)² - 16 + 8 = 0
Then:
(y + 4)² = -2x - 8
To make it clearer,
(y + 4)² = -2(x + 4)
This is now in the form of a sideway parabola with its vertex at (-4, -4). The parabola opens to the left because the x-term is negative. To sketch this graph, plot the vertex and a few points on either side of the parabola, knowing that when x is less than -4, the values of y will be real, and when x is greater than -4, there will be no real y-values because you cannot square a real number to get a negative number. So the graph is confined to the left side of the vertical line x = -4.