Step by step
The question
Question: draw the slope field of dy/dx = 2x − 2.
1 The idea
dy/dx is the slope of a solution curve.
For each point x, y, the equation tells us that slope.
So we draw a short line with that slope at each point: a slope field.
2 Slopes at some points
Put x and y of each point into 2x − 2.
At (0, 0): 2 × 0 − 2 = −2.
At (1, 0): 2 × 1 − 2 = 0.
Careful!
x comes first, then y.
At (1, 0), x is 1 and y is 0.
3 Draw the slope lines
Now the grid.
The slopes we found: a short purple line at each point.
Slope 0 is flat, slope 1 goes up like a hill, a minus slope goes down.
The same at every point of the grid.
This is the slope field.
4 Read the picture
The lines are flat where 2x − 2 = 0: on the upright line x = 1.
Every line in a column has the same slope: y does not change it.
Any solution curve follows these lines.
5 Check
Let's check one more point, (2, −1): the slope is 2.
Its line on the grid goes up, steeply.
Right!
Let's look back.
First: the idea.
Then: slopes at some points.
Then: draw the slope lines.
Then: read the picture.
Last: check.