Final answer:
To sketch the graph of the function f(x, y) = y² + 9(x-1)², the graph should be recognized as an ellipse centered at (1, 0) with vertices 3 units away in the x-direction and 1 unit away in the y-direction. An ellipse is then drawn connecting these vertices.
Step-by-step explanation:
Sketching the Graph of the Function f(x, y) = y² + 9(x-1)²
To sketch the graph of the function f(x, y) = y² + 9(x-1)², we recognize that this is an equation of an ellipse because it is in the general form Ax² + By² = C, where A and B are constants and both are positive. In this case, the ellipse will be centered at (1, 0), since the x-term is (x-1) and the y-term is y with no shift. The coefficient 9 for the (x-1)² term indicates that the stretch in the x-direction is sqrt(9) = 3, and since there is no coefficient for the y² term, the stretch in the y-direction is 1.
To plot this ellipse, one might start by plotting the center at (1, 0), then at a distance of 3 units to the left and right for the vertices along the x-axis, and 1 unit up and down for the vertices along the y-axis. Once these points are plotted, an ellipse can be drawn to connect these points smoothly. It's important to label the axes, scale them accordingly, and label the graph with f(x, y) as the function.