Final answer:
The function f(x, y) = y² + 9 is a parabolic cylinder independent of 'x'. To sketch its graph, compute and plot points on the yz-plane for various 'y' values, then draw the corresponding parabola, extending it in parallel along the entire x-axis.
Step-by-step explanation:
To sketch the graph of the function f(x, y) = y² + 9, you need to recognize that this equation represents a parabolic cylinder. Considering that the equation does not include 'x', it suggests that the parabola is parallel to the x-axis. For any value of 'x', the 'y' values produce a parabola that opens upwards with its vertex at (0, -9) in the yz-plane.
In order to plot the graph, choose various values for 'y' and calculate the corresponding 'z' values. This exercise is similar to plotting y = 9 + 3x, where you select different values of 'x' to find 'y'. However, since our function is independent of 'x', the process is even simpler. Let's set x constant and find some points for 'y' and 'z':
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- If y = 0, then z = 0² + 9 = 9.
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- If y = 1, then z = 1² + 9 = 10.
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- If y = -1, then z = (-1)² + 9 = 10.
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- If y = 2, then z = 2² + 9 = 13.
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- If y = -2, then z = (-2)² + 9 = 13.
Once you have a set of points for 'y' and 'z', you can plot them on the yz-plane. Then, simply draw a smooth curve that represents the parabola for those points. Since the function is independent of 'x', this parabola will be consistent across all values of 'x', making it a parabolic cylinder.