See how all of these slopes are independent of the x-values, so for each value of y, the slope is the same
horizontally, but is different vertically.
Now let’s do a slightly harder example.
Example 3: Given = xy, sketch the slope field of the function.
Now, we have to think about both the x- and y-values at each point. Let’s calculate a few slopes.
At (0, 0), the slope is (0)(0) = 0.At (1, 0), the slope is (1)(0) = 0.At (2, 0), the slope is (2)(0) = 0.At (0, 1), the slope is (0)(1) = 0.At (0, 2), the slope is (0)(2) = 0.So the slope will be zero at any point on the coordinate axes.
At (1, 1), the slope is (1)(1) = 1.At (1, 2), the slope is (1)(2) = 2.At (1, −1), the slope is (1)(−1) = −1.At (1, −2), the slope is (1)(−2) = −2.So the slope at any point where x = 1 will be the y-value. Similarly, you should see that the slope at any
point where y = 1 will be the x-value. As we move out the coordinate axes, slopes will get steeper—
whether positive or negative.