Final answer:
Level curves for the function f(x, y) = y - 2x² at c = 2 can be sketched by setting f(x, y) to 2, giving the equation y = 2x² + 2, and plotting its data pairs to form a parabola. The curve will cross the y-axis at y = 2, so we can calculate points by substituting values for x and then sketch the parabola.
Step-by-step explanation:
To sketch the level curves of the function f(x, y) = y - 2x² for c = 2, we set f(x, y) equal to 2 and solve for y:
2 = y - 2x²
y = 2x² + 2
This equation represents a family of parabolas for different values of c. Each level curve is a parabola opening upwards with a vertex at the point (0, c). For c = 2, the curve will be a parabola that crosses the y-axis at y = 2. To obtain various points on this parabola, we can plug in different x values into the equation y = 2x² + 2 and plot these data pairs on a graph. For example, if we plug in x = -1, 0, and 1, we'll get data pairs (-1,4), (0,2), and (1,4) respectively. Plotting these, along with additional points, will give us the shape of the curve.
Keep in mind that the level curve for c = 2 is just one of the many curves on a graph that would represent different values of the constant c. Together, they form a set of curves that indicate the height of the function above the x-y plane.